<div id="pf1" class="pf w0 h0" data-page-no="1"><div class="pc pc1 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://files.passeidireto.com/e1597c04-9ff2-4637-afbf-78fed828f3b2/bg1.png"><div class="c x0 y1 w2 h2"><div class="t m0 x1 h3 y2 ff1 fs0 fc0 sc0 ls0 ws0">EA<span class="_0 blank"></span>E0<span class="_0 blank"></span>207<span class="_0 blank"></span>:<span class="_1 blank"> </span>M<span class="_0 blank"></span>ate<span class="_0 blank"></span>mát<span class="_0 blank"></span>ica A<span class="_0 blank"></span>pli<span class="_0 blank"></span>cad<span class="_0 blank"></span>a à Eco<span class="_0 blank"></span>no<span class="_0 blank"></span>mia</div><div class="t m0 x2 h4 y3 ff1 fs1 fc0 sc0 ls0 ws1">Aula<span class="_2 blank"> </span>7:<span class="_3 blank"> </span>Indep endência<span class="_2 blank"> </span>Linea<span class="_0 blank"></span>r</div><div class="t m0 x3 h4 y4 ff1 fs1 fc1 sc0 ls0 ws2">Ma<span class="_0 blank"></span>rcos Y. Nakaguma</div><div class="t m0 x4 h4 y5 ff1 fs1 fc1 sc0 ls0 ws3">23/08/2017</div><div class="t m0 x5 h5 y6 ff1 fs2 fc1 sc0 ls0">1</div></div><div class="c x0 y7 w2 h2"><div class="t m0 x6 h6 y8 ff1 fs3 fc1 sc0 ls0 ws4">I<span class="_0 blank"></span>n<span class="_0 blank"></span>d<span class="_0 blank"></span>e<span class="_0 blank"></span>pe<span class="_0 blank"></span>n<span class="_0 blank"></span>d<span class="_0 blank"></span>ê<span class="_0 blank"></span>n<span class="_4 blank"></span>c<span class="_0 blank"></span>ia L<span class="_4 blank"></span>i<span class="_0 blank"></span>n<span class="_0 blank"></span>e<span class="_0 blank"></span>a<span class="_4 blank"></span>r</div><div class="t m0 x5 h5 y9 ff1 fs2 fc1 sc0 ls0">2</div></div><div class="c x0 ya w2 h2"><div class="t m0 x7 h3 yb ff1 fs0 fc0 sc0 ls0 ws5">In<span class="_0 blank"></span>trodu<span class="_0 blank"></span>ção</div><div class="t m0 x8 h4 yc ff1 fs1 fc1 sc0 ls0 ws6">Nesta pa<span class="_0 blank"></span>rte do curso, continuaremos o nosso estudo sob<span class="_0 blank"></span>re sistemas de</div><div class="t m0 x8 h4 yd ff1 fs1 fc1 sc0 ls0 ws2">equações linea<span class="_0 blank"></span>res.<span class="_3 blank"> </span>Já sab<span class="_5 blank"> </span>emos como "determina<span class="_0 blank"></span>r" quando um</div><div class="t m0 x8 h4 ye ff1 fs1 fc1 sc0 ls0 ws7">sistema p<span class="_5 blank"> </span>ossui solução e quando esta solução é única.</div><div class="t m0 x8 h4 yf ff1 fs1 fc1 sc0 ls0 ws8">P<span class="_0 blank"></span>orém, quando existem in\u2026nitas soluções, gosta<span class="_0 blank"></span>ríamos de sab<span class="_5 blank"> </span>er dizer</div><div class="t m0 x8 h4 y10 ff1 fs1 fc1 sc0 ls0 ws2">algo sob<span class="_0 blank"></span>re o "tamanho" deste conjunto solução.</div><div class="t m0 x8 h4 y11 ff1 fs1 fc1 sc0 ls0 ws2">V<span class="_0 blank"></span>eremos que, neste caso, o "tamanho" de uma conjunto de soluções</div><div class="t m0 x8 h4 y12 ff1 fs1 fc1 sc0 ls0 ws8">p<span class="_5 blank"> </span>ode ser capturado p<span class="_5 blank"> </span>ela sua <span class="fc2 ws3 v0">dimensão<span class="fc1 ws9">.<span class="_3 blank"> </span>O<span class="_2 blank"> </span>conceito<span class="_2 blank"> </span>de<span class="_2 blank"> </span>dimensão,<span class="_2 blank"> </span>por</span></span></div><div class="t m0 x8 h4 y13 ff1 fs1 fc1 sc0 ls0 wsa">sua vez, está intimamente relacionado à idéia de <span class="fc2 wsb">indep endência<span class="_2 blank"> </span>linea<span class="_0 blank"></span>r<span class="fc1">.</span></span></div><div class="t m0 x5 h5 y14 ff1 fs2 fc1 sc0 ls0">3</div></div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div> <div id="pf2" class="pf w0 h0" data-page-no="2"><div class="pc pc2 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://files.passeidireto.com/e1597c04-9ff2-4637-afbf-78fed828f3b2/bg2.png"><div class="c x0 y1 w2 h2"><div class="t m0 x7 h3 y15 ff1 fs0 fc0 sc0 ls0 wsd">In<span class="_0 blank"></span>depen<span class="_0 blank"></span>dên<span class="_0 blank"></span>cia<span class="_6 blank"> </span>Li<span class="_0 blank"></span>nea<span class="_4 blank"></span>r:<span class="_1 blank"> </span>M<span class="_4 blank"></span>otiv<span class="_0 blank"></span>açã<span class="_0 blank"></span>o</div><div class="t m0 x8 h4 y16 ff1 fs1 fc1 sc0 ls0 ws8">Considere o seguinte sistema de equações linea<span class="_0 blank"></span>res:</div><div class="t m0 x9 h7 y17 ff2 fs1 fc1 sc0 ls0">8</div><div class="t m0 x9 h7 y18 ff2 fs1 fc1 sc0 ls0"><</div><div class="t m0 x9 h7 y19 ff2 fs1 fc1 sc0 ls0">:</div><div class="t m0 xa h8 y1a ff1 fs1 fc1 sc0 ls0 ws3">2<span class="ff3 ls1">x</span><span class="fs2 ls2 v1">1</span><span class="ff4 fs4 ls3 v0">+</span><span class="v0">2<span class="ff3 ls4">x</span><span class="fs2 ls5 v1">2</span><span class="ff5 fs4 ls6">\ue000</span><span class="ff3">x</span><span class="fs2 ls7 v1">3</span><span class="ff4 fs4 ls8">=</span>-3</span></div><div class="t m0 xa h9 y1b ff1 fs1 fc1 sc0 ls0 ws3">4<span class="ff3 ls1">x</span><span class="fs2 ls9 v1">1</span><span class="ff4 fs4 lsa v0">+</span><span class="v0">2<span class="ff3">x</span><span class="fs2 lsb v1">3</span><span class="ff4 fs4 lsc">=</span>8</span></div><div class="t m0 xb h9 y1c ff1 fs1 fc1 sc0 ls0 ws3">6<span class="ff3 ls1">x</span><span class="fs2 lsd v1">2</span><span class="ff5 fs4 lsa v0">\ue000</span><span class="v0">3<span class="ff3">x</span><span class="fs2 lsb v1">3</span><span class="ff4 fs4 ls3">=</span>-12</span></div><div class="t m0 x8 h4 y1d ff1 fs1 fc1 sc0 ls0 ws2">Note que este sistema p<span class="_5 blank"> </span>ode ser re-expresso em termos de veto<span class="_0 blank"></span>res:</div><div class="t m0 xc ha y1e ff3 fs1 fc1 sc0 ls1">x<span class="ff1 fs2 lse v1">1</span><span class="ff2 ls0 v2">0</span></div><div class="t m0 xd h7 y1f ff2 fs1 fc1 sc0 ls0">@</div><div class="t m0 xe h4 y20 ff1 fs1 fc1 sc0 ls0">2</div><div class="t m0 xe h4 y21 ff1 fs1 fc1 sc0 ls0">4</div><div class="t m0 xe h4 y22 ff1 fs1 fc1 sc0 ls0">0</div><div class="t m0 x9 h7 y23 ff2 fs1 fc1 sc0 ls0">1</div><div class="t m0 x9 hb y1f ff2 fs1 fc1 sc0 lsf">A<span class="ff4 fs4 ls10 v3">+</span><span class="ff3 ls0 ws3 v3">x</span><span class="ff1 fs2 lse v4">2</span><span class="ls0 v5">0</span></div><div class="t m0 xf h7 y24 ff2 fs1 fc1 sc0 ls0">@</div><div class="t m0 x10 h4 y25 ff1 fs1 fc1 sc0 ls0">2</div><div class="t m0 x10 h4 y26 ff1 fs1 fc1 sc0 ls0">0</div><div class="t m0 x10 h4 y27 ff1 fs1 fc1 sc0 ls0">6</div><div class="t m0 x11 h7 y28 ff2 fs1 fc1 sc0 ls0">1</div><div class="t m0 x11 hc y24 ff2 fs1 fc1 sc0 ls11">A<span class="ff4 fs4 ls10 v3">+</span><span class="ff3 ls1 v3">x</span><span class="ff1 fs2 ls12 v4">3</span><span class="ls0 v5">0</span></div><div class="t m0 x12 h7 y24 ff2 fs1 fc1 sc0 ls0">@</div><div class="t m0 x13 h4 y25 ff1 fs1 fc1 sc0 ls0">1</div><div class="t m0 x13 h4 y26 ff1 fs1 fc1 sc0 ls0">2</div><div class="t m0 x13 h4 y27 ff1 fs1 fc1 sc0 ls0">3</div><div class="t m0 x14 h7 y28 ff2 fs1 fc1 sc0 ls0">1</div><div class="t m0 x14 hd y24 ff2 fs1 fc1 sc0 ls13">A<span class="ff4 fs4 ls14 v3">=</span><span class="ls0 v5">0</span></div><div class="t m0 x15 h7 y29 ff2 fs1 fc1 sc0 ls0">@</div><div class="t m0 x16 h4 y2a ff1 fs1 fc1 sc0 ls0">-3</div><div class="t m0 x17 h4 y2b ff1 fs1 fc1 sc0 ls0">8</div><div class="t m0 x18 h4 y2c ff1 fs1 fc1 sc0 ls0">-12</div><div class="t m0 x19 h7 y2d ff2 fs1 fc1 sc0 ls0">1</div><div class="t m0 x19 h7 y29 ff2 fs1 fc1 sc0 ls15">A<span class="ff5 fs4 ls0 v3">$</span></div><div class="t m0 x1a he y2e ff3 fs1 fc1 sc0 ls0 ws3">x<span class="ff1 fs2 ls16 v1">1</span><span class="ff6">a<span class="ff1 fs2 ls17 v1">1</span><span class="ff4 fs4 ls10">+</span></span><span class="ls1">x<span class="ff1 fs2 ls18 v1">2</span></span><span class="ff6">a<span class="ff1 fs2 ls19 v1">2</span><span class="ff4 fs4 ls1a">+</span></span><span class="ls1">x<span class="ff1 fs2 ls16 v1">3</span></span><span class="ff6">a<span class="ff1 fs2 ls1b v1">3</span><span class="ff4 fs4 ls1c">=</span>b</span></div><div class="t m0 x8 h4 y2f ff1 fs1 fc1 sc0 ls0 ws2">Assim, o sistema p<span class="_5 blank"> </span>ossui solução se, e somente se, <span class="ff6 ls1d">b</span><span class="wse">puder ser escrito</span></div><div class="t m0 x8 h4 y30 ff1 fs1 fc1 sc0 ls0 wsf">como uma <span class="fc2 ws10 v0">combinação linea<span class="_0 blank"></span>r <span class="fc1 ws11">dos vetores coluna da matriz de</span></span></div><div class="t m0 x8 h4 y31 ff1 fs1 fc1 sc0 ls0 ws12">co e\u2026cientes<span class="_2 blank"> </span><span class="ff3 ws3">A<span class="ff7">.</span></span></div><div class="t m0 x5 h5 y32 ff1 fs2 fc1 sc0 ls0">4</div></div><div class="c x0 y7 w2 h2"><div class="t m0 x7 h3 y33 ff1 fs0 fc0 sc0 ls0 wsd">In<span class="_0 blank"></span>depen<span class="_0 blank"></span>dên<span class="_0 blank"></span>cia<span class="_6 blank"> </span>Li<span class="_0 blank"></span>nea<span class="_4 blank"></span>r:<span class="_1 blank"> </span>M<span class="_4 blank"></span>otiv<span class="_0 blank"></span>açã<span class="_0 blank"></span>o</div><div class="t m0 x8 hf y34 ff1 fs1 fc1 sc0 ls0 ws7">Como a única solução desse sistema é dada p<span class="_5 blank"> </span>o<span class="_0 blank"></span>r <span class="ff3 ws3">x</span><span class="fs2 ls1e v1">1</span><span class="ff4 fs4 ls1f">=</span><span class="fs2 v6">1</span></div><div class="t m0 x1b h10 y35 ff1 fs2 fc1 sc0 ls20">2<span class="ff7 fs1 ls21 v7">,<span class="ff3 ls1">x</span></span><span class="ls22 v8">2</span><span class="ff4 fs4 ls14 v7">=<span class="ff5 ls23">\ue000</span></span><span class="ls0 v9">1</span></div><div class="t m0 x1c h11 y35 ff1 fs2 fc1 sc0 ls24">2<span class="fs1 ls0 v7">e</span></div><div class="t m0 x8 h12 y36 ff3 fs1 fc1 sc0 ls1">x<span class="ff1 fs2 ls25 v1">3</span><span class="ff4 fs4 ls1c v0">=<span class="ff1 fs1 ls0 ws7">3, temos que:</span></span></div><div class="t m0 x1d h7 y37 ff2 fs1 fc1 sc0 ls0">0</div><div class="t m0 x1d h7 y38 ff2 fs1 fc1 sc0 ls0">@</div><div class="t m0 xe h4 y39 ff1 fs1 fc1 sc0 ls0">-3</div><div class="t m0 x1e h4 y3a ff1 fs1 fc1 sc0 ls0">8</div><div class="t m0 x1f h4 y3b ff1 fs1 fc1 sc0 ls0">-12</div><div class="t m0 x2 h7 y37 ff2 fs1 fc1 sc0 ls0">1</div><div class="t m0 x2 h13 y38 ff2 fs1 fc1 sc0 ls26">A<span class="ff4 fs4 ls27 v3">=</span><span class="ff1 ls0 va">1</span></div><div class="t m0 x20 h14 y3c ff1 fs1 fc1 sc0 ls28">2<span class="ff2 ls0 vb">0</span></div><div class="t m0 x21 h7 y3d ff2 fs1 fc1 sc0 ls0">@</div><div class="t m0 x22 h4 y3e ff1 fs1 fc1 sc0 ls0">2</div><div class="t m0 x22 h4 y3f ff1 fs1 fc1 sc0 ls0">4</div><div class="t m0 x22 h4 y40 ff1 fs1 fc1 sc0 ls0">0</div><div class="t m0 x23 h7 y41 ff2 fs1 fc1 sc0 ls0">1</div><div class="t m0 x23 h15 y42 ff2 fs1 fc1 sc0 ls29">A<span class="ff5 fs4 ls2a v3">\ue000</span><span class="ff1 ls0 va">1</span></div><div class="t m0 x24 h14 y3c ff1 fs1 fc1 sc0 ls2b">2<span class="ff2 ls0 vb">0</span></div><div class="t m0 x25 h7 y3d ff2 fs1 fc1 sc0 ls0">@</div><div class="t m0 x26 h4 y3e ff1 fs1 fc1 sc0 ls0">2</div><div class="t m0 x26 h4 y3f ff1 fs1 fc1 sc0 ls0">0</div><div class="t m0 x26 h4 y40 ff1 fs1 fc1 sc0 ls0">6</div><div class="t m0 x27 h7 y41 ff2 fs1 fc1 sc0 ls0">1</div><div class="t m0 x27 h16 y42 ff2 fs1 fc1 sc0 ls2c">A<span class="ff4 fs4 ls10 v3">+</span><span class="ff1 ls2d v3">3</span><span class="ls0 v5">0</span></div><div class="t m0 x28 h7 y43 ff2 fs1 fc1 sc0 ls0">@</div><div class="t m0 x1b h4 y44 ff1 fs1 fc1 sc0 ls0">1</div><div class="t m0 x1b h4 y45 ff1 fs1 fc1 sc0 ls0">2</div><div class="t m0 x1b h4 y46 ff1 fs1 fc1 sc0 ls0">3</div><div class="t m0 x19 h7 y47 ff2 fs1 fc1 sc0 ls0">1</div><div class="t m0 x19 h7 y48 ff2 fs1 fc1 sc0 ls2e">A<span class="ff7 ls0 v3">,</span></div><div class="t m0 x8 h4 y49 ff1 fs1 fc1 sc0 ls0 ws13">ou seja,</div><div class="t m0 xf h17 y4a ff6 fs1 fc1 sc0 ls2f">b<span class="ff4 fs4 ls23 v0">=</span><span class="ff1 ls0 vc">1</span></div><div class="t m0 x29 h18 y4b ff1 fs1 fc1 sc0 ls30">2<span class="ff6 ls0 ws3 vc">a</span><span class="fs2 ls31 vd">1</span><span class="ff5 fs4 ls2a vc">\ue000</span><span class="ls0 ve">1</span></div><div class="t m0 x2a h19 y4b ff1 fs1 fc1 sc0 ls32">2<span class="ff6 ls0 ws3 vc">a</span><span class="fs2 ls33 vd">2</span><span class="ff4 fs4 ls34 vc">+</span><span class="ls0 ws3 vc">3<span class="ff6">a<span class="ff1 fs2 v1">3</span></span></span></div><div class="t m0 x8 h4 y4c ff1 fs1 fc1 sc0 ls0 ws14">Neste caso, dizemos que <span class="ff6 ls35">b</span><span class="ls36">é</span><span class="fc2 ws12 v0">linea<span class="_0 blank"></span>rmente<span class="_2 blank"> </span>dep endente<span class="_2 blank"> </span><span class="fc1 ws15">dos vetores <span class="ff6 ws3">a</span><span class="fs2 ls37 v1">1</span>,</span></span></div><div class="t m0 x8 h4 y4d ff6 fs1 fc1 sc0 ls0 ws3">a<span class="ff1 fs2 ls38 v1">2</span><span class="ff1 ls39">e</span>a<span class="ff1 fs2 ls3a v1">3</span><span class="ff7">.</span></div><div class="t m0 x5 h5 y4e ff1 fs2 fc1 sc0 ls0">5</div></div><div class="c x0 ya w2 h2"><div class="t m0 x7 h3 yb ff1 fs0 fc0 sc0 ls0 wsd">In<span class="_0 blank"></span>depen<span class="_0 blank"></span>dên<span class="_0 blank"></span>cia<span class="_6 blank"> </span>Li<span class="_0 blank"></span>nea<span class="_4 blank"></span>r:<span class="_1 blank"> </span>D<span class="_4 blank"></span>e\u2026n<span class="_0 blank"></span>içõe<span class="_0 blank"></span>s</div><div class="t m0 x8 h1a y4f ff1 fs1 fc2 sc0 ls0 ws16">De\u2026nição: <span class="fc1 ws17">Seja <span class="ff6 ws3">v</span><span class="fs2 ls18 v1">1</span><span class="ff7 ls3b">,</span><span class="ff6 ws3">v</span><span class="fs2 ls18 v1">2</span><span class="ff7 ws18">, ...,<span class="_2 blank"> </span><span class="ff6 ws3">v<span class="ff3 fs2 ls3c v1">k</span><span class="ff5 fs4 ls3d">2</span><span class="ff8 ls3e">R<span class="ff3 fs2 ls3f v7">n</span></span></span></span><span class="ls40">e<span class="ff3 ls41">a</span><span class="fs2 ls18 v1">1</span><span class="ls21">,</span></span><span class="ff7 ws19">..., <span class="ff3 ls42">a<span class="fs2 ls43 v1">k</span><span class="ff5 fs4 ls44">2</span><span class="ff8 ls45">R</span></span><span class="ls3b">.</span></span>Então,</span></div><div class="t m0 x2b h1b y50 ff3 fs1 fc1 sc0 ls42">a<span class="ff1 fs2 ls16 v1">1</span><span class="ff7 ls0 ws3">.<span class="ff6">v<span class="ff1 fs2 ls46 v1">1</span><span class="ff4 fs4 ls10 v0">+</span></span></span><span class="ls41 v0">a<span class="ff1 fs2 ls16 v1">2</span><span class="ff7 ls0 ws3">.<span class="ff6">v<span class="ff1 fs2 ls47 v1">2</span><span class="ff4 fs4 ls10">+</span></span><span class="ws1a">... <span class="ff4 fs4 ls10">+</span></span></span><span class="ls42">a<span class="fs2 ls48 v1">k</span><span class="ff7 ls0 ws3">.<span class="ff6">v</span></span><span class="fs2 ls49 v1">k</span><span class="ff5 fs4 ls4a">2</span><span class="ff8 ls4b">R</span><span class="fs2 ls0 v6">n</span></span></span></div><div class="t m0 x8 h1c y51 ff1 fs1 fc1 sc0 ls0 ws1b">é chamada <span class="fc2 ws1c v0">combinação linea<span class="_0 blank"></span>r <span class="fc1 ws1d">de <span class="ff6 ws3">v</span><span class="fs2 ls3a v1">1</span><span class="ff7 ws2">, ..., <span class="ff6 ws3">v<span class="ff3 fs2 ls4c v1">k</span></span>.</span></span></span></div><div class="t m0 x8 h4 y52 ff1 fs1 fc2 sc0 ls0 ws3">De\u2026nição:</div><div class="t m0 x2c h1d y53 ff3 fs5 fc0 sc0 ls4d">i<span class="ff7 ls4e">.<span class="ff1 fc1 ls0 ws1e">O conjunto de veto<span class="_0 blank"></span>res <span class="ff6 ws1f">v</span><span class="fs2 ls16 v1">1</span><span class="ff7 ls4f">,</span><span class="ff6 ws1f">v</span><span class="fs2 ls18 v1">2</span><span class="ff7 ws20">, ...,<span class="_2 blank"> </span><span class="ff6 ws1f">v<span class="ff3 fs2 ls50 v3">k</span></span></span><span class="ws21">em <span class="ff8 ls51">R<span class="ff3 fs2 ls52 v7">n</span></span><span class="ls53">é</span><span class="fc2 ws22">linea<span class="_0 blank"></span>rmente<span class="_2 blank"> </span>dependente</span></span></span></span></div><div class="t m0 x2d h1e y54 ff1 fs5 fc1 sc0 ls0 ws23">se, e somente se, p<span class="_5 blank"> </span>elo menos um dos veto<span class="_0 blank"></span>res <span class="ff6 ws1f">v</span><span class="fs2 ls16 v1">1</span><span class="ff7 ls54">,<span class="ff6 ls55">v</span></span><span class="fs2 ls18 v1">2</span><span class="ff7 ws20">, ...,<span class="_7 blank"> </span><span class="ff6 ws1f">v<span class="ff3 fs2 ls56 v3">k</span></span></span><span class="ws24">puder ser</span></div><div class="t m0 x2d h1e y55 ff1 fs5 fc1 sc0 ls0 ws1e">exp<span class="_0 blank"></span>resso como uma combinação linear dos demais.</div><div class="t m0 x2e h1e y56 ff3 fs5 fc0 sc0 ls0 ws25">ii <span class="ff7 ls57">.</span><span class="ff1 fc1 ws1e">O conjunto de veto<span class="_0 blank"></span>res <span class="ff6 ws1f">v</span><span class="fs2 ls16 v1">1</span><span class="ff7 ls4f">,</span><span class="ff6 ws1f">v</span><span class="fs2 ls18 v1">2</span><span class="ff7 ws20">, ...,<span class="_7 blank"> </span><span class="ff6 ws1f">v<span class="ff3 fs2 ls50 v3">k</span></span></span><span class="ls58">é</span><span class="fc2 ws26">linearmente<span class="_7 blank"> </span>independente<span class="_7 blank"> </span></span><span class="ws27">se, e</span></span></div><div class="t m0 x2d h1e y57 ff1 fs5 fc1 sc0 ls0 ws28">somente se, nenhum dos veto<span class="_0 blank"></span>res <span class="ff6 ws1f">v</span><span class="fs2 ls16 v1">1</span><span class="ff7 ls59">,</span><span class="ff6 ws1f">v</span><span class="fs2 ls16 v1">2</span><span class="ff7 ws29">, ...,<span class="_7 blank"> </span><span class="ff6 ws1f">v<span class="ff3 fs2 ls5a v3">k</span></span></span><span class="ws24">puder ser expresso como</span></div><div class="t m0 x2d h1e y58 ff1 fs5 fc1 sc0 ls0 ws2a">combinação linea<span class="_0 blank"></span>r dos demais.</div><div class="t m0 x5 h5 y59 ff1 fs2 fc1 sc0 ls0">6</div></div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div> <div id="pf3" class="pf w0 h0" data-page-no="3"><div class="pc pc3 w0 h0"><img fetchpriority="low" loading="lazy" class="bi x0 y0 w1 h1" alt="" src="https://files.passeidireto.com/e1597c04-9ff2-4637-afbf-78fed828f3b2/bg3.png"><div class="c x0 y1 w2 h2"><div class="t m0 x7 h3 y15 ff1 fs0 fc0 sc0 ls0 wsd">In<span class="_0 blank"></span>depen<span class="_0 blank"></span>dên<span class="_0 blank"></span>cia<span class="_6 blank"> </span>Li<span class="_0 blank"></span>nea<span class="_4 blank"></span>r:<span class="_1 blank"> </span>D<span class="_4 blank"></span>e\u2026n<span class="_0 blank"></span>içõe<span class="_0 blank"></span>s</div><div class="t m0 x8 h4 y5a ff1 fs1 fc1 sc0 ls0 ws2b">Uma de\u2026nição equivalente de dep<span class="_5 blank"> </span>end<span class="_0 blank"></span>ência linear é a seguinte:</div><div class="t m0 x8 h4 y5b ff1 fs1 fc2 sc0 ls0 ws3">De\u2026nição:</div><div class="t m0 x2c h1f y5c ff3 fs5 fc0 sc0 ls4d">i<span class="ff7 ls4e">.<span class="ff1 fc1 ls0 ws1e">O conjunto de veto<span class="_0 blank"></span>res <span class="ff6 ws1f">v</span><span class="fs2 ls16 v1">1</span><span class="ff7 ls4f">,</span><span class="ff6 ws1f">v</span><span class="fs2 ls18 v1">2</span><span class="ff7 ws29">, ...,<span class="_7 blank"> </span><span class="ff6 ws1f">v<span class="ff3 fs2 ls50 v3">k</span></span></span><span class="ws21">em <span class="ff8 ls51">R<span class="ff3 fs2 ls52 v7">n</span></span><span class="ls53">é</span><span class="fc2 ws22">linearmente<span class="_7 blank"> </span>dependente</span></span></span></span></div><div class="t m0 x2d h1e y5d ff1 fs5 fc1 sc0 ls0 ws2c">se, e somente se, existirem escalares <span class="ff3 ls5b">c</span><span class="fs2 ls5c v1">1</span><span class="ff7 ls5d">,<span class="ff3 ls5e">c</span></span><span class="fs2 ls5f v1">2<span class="ff9 ls60">,</span></span><span class="ff7 ws2d">..., <span class="ff3 ls61">c<span class="fs2 ls5a v3">k</span><span class="ls0 ws2e">não<span class="_7 blank"> </span>todos<span class="_7 blank"> </span>zero</span></span></span><span class="ws28">, tais</span></div><div class="t m0 x2d h1e y5e ff1 fs5 fc1 sc0 ls0 ws1f">que:</div><div class="t m0 x1a h20 y5f ff3 fs5 fc1 sc0 ls61">c<span class="ff1 fs2 ls18 v1">1</span><span class="ff7 ls0 ws1f">.<span class="ff6">v<span class="ff1 fs2 ls62 v1">1</span><span class="ff4 fs6 ls63">+</span></span></span><span class="ls5b">c<span class="ff1 fs2 ls3a v1">2</span><span class="ff7 ls0 ws1f">.<span class="ff6">v<span class="ff1 fs2 ls62 v1">2</span><span class="ff4 fs6 ls64">+</span></span><span class="ws2f">... <span class="ff4 fs6 ls65">+</span></span></span></span>c<span class="fs2 ls48 v3">k</span><span class="ff7 ls0 ws1f">.<span class="ff6">v</span></span><span class="fs2 ls66 v3">k</span><span class="ff4 fs6 ls67">=</span><span class="ff6 ls0">0</span></div><div class="t m0 x2e h1e y60 ff3 fs5 fc0 sc0 ls0 ws25">ii <span class="ff7 ls57">.</span><span class="ff1 fc1 ws1e">O conjunto de veto<span class="_0 blank"></span>res <span class="ff6 ws1f">v</span><span class="fs2 ls16 v1">1</span><span class="ff7 ls4f">,</span><span class="ff6 ws1f">v</span><span class="fs2 ls18 v1">2</span><span class="ff7 ws20">, ...,<span class="_7 blank"> </span><span class="ff6 ws1f">v<span class="ff3 fs2 ls50 v3">k</span></span></span><span class="ls58">é</span><span class="fc2 ws30">linearmente<span class="_7 blank"> </span>independente<span class="_7 blank"> </span></span><span class="ws2c">se, e</span></span></div><div class="t m0 x2d h20 y61 ff1 fs5 fc1 sc0 ls0 ws28">somente se, a equação <span class="ff3 ls61">c</span><span class="fs2 ls16 v1">1</span><span class="ff7 ws1f">.<span class="ff6">v</span></span><span class="fs2 ls68 v1">1</span><span class="ff4 fs6 ls69">+</span><span class="ff3 ls61">c</span><span class="fs2 ls16 v1">2</span><span class="ff7 ws1f">.<span class="ff6">v</span></span><span class="fs2 ls6a v1">2</span><span class="ff4 fs6 ls6b">+</span><span class="ff7 ws31">...<span class="ff4 fs6 ls69">+</span><span class="ff3 ls61">c<span class="fs2 ls48 v3">k</span></span><span class="ws1f">.<span class="ff6">v<span class="ff3 fs2 ls6c v3">k</span><span class="ff4 fs6 ls6d">=</span><span class="ls6e">0</span></span></span></span><span class="ws32">é satisfeita</span></div><div class="t m0 x2d h20 y62 ff3 fs5 fc1 sc0 ls0 ws33">somente <span class="ff1 ws34">quando </span><span class="ls6f">c<span class="ff1 fs2 ls70 v1">1</span><span class="ff4 fs6 ls6d">=</span></span><span class="ff7 ws35">... <span class="ff4 fs6 ls6d">=</span></span><span class="ls6f">c<span class="fs2 ls71 v3">k</span><span class="ff4 fs6 ls6d">=</span></span><span class="ff1 ws1f">0<span class="ff7">.</span></span></div><div class="t m0 x5 h5 y32 ff1 fs2 fc1 sc0 ls0">7</div></div><div class="c x0 y7 w2 h2"><div class="t m0 x7 h3 y33 ff1 fs0 fc0 sc0 ls0 wsd">In<span class="_0 blank"></span>depen<span class="_0 blank"></span>dên<span class="_0 blank"></span>cia<span class="_6 blank"> </span>Li<span class="_0 blank"></span>nea<span class="_4 blank"></span>r:<span class="_1 blank"> </span>E<span class="_0 blank"></span>xe<span class="_0 blank"></span>mp<span class="_0 blank"></span>los</div><div class="t m0 x8 h4 y63 ff1 fs1 fc2 sc0 ls0 ws7">Exemplo 1:<span class="_3 blank"> </span><span class="fc1 ws8">Os veto<span class="_0 blank"></span>res</span></div><div class="t m0 x2f h21 y64 ff6 fs1 fc1 sc0 ls72">e<span class="ff1 fs2 ls73 v1">1</span><span class="ff4 fs4 ls14 v0">=</span><span class="ff2 ls0 v2">0</span></div><div class="t m0 x30 h7 y65 ff2 fs1 fc1 sc0 ls0">@</div><div class="t m0 x31 h4 y66 ff1 fs1 fc1 sc0 ls0">1</div><div class="t m0 x31 h4 y67 ff1 fs1 fc1 sc0 ls0">0</div><div class="t m0 x31 h4 y68 ff1 fs1 fc1 sc0 ls0">0</div><div class="t m0 x1a h7 y69 ff2 fs1 fc1 sc0 ls0">1</div><div class="t m0 x1a h22 y65 ff2 fs1 fc1 sc0 ls2e">A<span class="ff7 ls74 v3">,<span class="ff6 ls0 ws3">e<span class="ff1 fs2 ls75 v1">2</span><span class="ff4 fs4 ls14 v0">=</span><span class="ff2 v2">0</span></span></span></div><div class="t m0 x32 h7 y65 ff2 fs1 fc1 sc0 ls0">@</div><div class="t m0 x33 h4 y66 ff1 fs1 fc1 sc0 ls0">0</div><div class="t m0 x33 h4 y67 ff1 fs1 fc1 sc0 ls0">1</div><div class="t m0 x33 h4 y68 ff1 fs1 fc1 sc0 ls0">0</div><div class="t m0 x34 h7 y69 ff2 fs1 fc1 sc0 ls0">1</div><div class="t m0 x34 h22 y65 ff2 fs1 fc1 sc0 ls76">A<span class="ff7 ls77 v3">,<span class="ff6 ls0 ws3">e<span class="ff1 fs2 ls78 v1">3</span><span class="ff4 fs4 ls79 v0">=</span><span class="ff2 v2">0</span></span></span></div><div class="t m0 x28 h7 y65 ff2 fs1 fc1 sc0 ls0">@</div><div class="t m0 x35 h4 y66 ff1 fs1 fc1 sc0 ls0">0</div><div class="t m0 x35 h4 y67 ff1 fs1 fc1 sc0 ls0">0</div><div class="t m0 x35 h4 y68 ff1 fs1 fc1 sc0 ls0">1</div><div class="t m0 x19 h7 y69 ff2 fs1 fc1 sc0 ls0">1</div><div class="t m0 x19 h7 y65 ff2 fs1 fc1 sc0 ls0">A</div><div class="t m0 x8 h23 y6a ff1 fs1 fc1 sc0 ls0 ws36">em <span class="ff8 ls3e">R</span><span class="fs2 ls38 v7">3</span><span class="ws9">são<span class="_2 blank"> </span>linea<span class="_0 blank"></span>rmente<span class="_2 blank"> </span>indep endentes<span class="_2 blank"> </span>p ois:</span></div><div class="t m0 x8 h24 y6b ff3 fs1 fc1 sc0 ls7a">c<span class="ff1 fs2 ls7b v1">1</span><span class="ff2 ls0 v2">0</span></div><div class="t m0 x36 h7 y6c ff2 fs1 fc1 sc0 ls0">@</div><div class="t m0 x37 h4 y6d ff1 fs1 fc1 sc0 ls0">1</div><div class="t m0 x37 h4 y6e ff1 fs1 fc1 sc0 ls0">0</div><div class="t m0 x37 h4 y6f ff1 fs1 fc1 sc0 ls0">0</div><div class="t m0 x38 h7 y70 ff2 fs1 fc1 sc0 ls0">1</div><div class="t m0 x38 h25 y6c ff2 fs1 fc1 sc0 ls7c">A<span class="ff4 fs4 ls7d v3">+</span><span class="ff3 ls7a v3">c</span><span class="ff1 fs2 ls7b v4">2</span><span class="ls0 v5">0</span></div><div class="t m0 x39 h7 y71 ff2 fs1 fc1 sc0 ls0">@</div><div class="t m0 x2b h4 y72 ff1 fs1 fc1 sc0 ls0">0</div><div class="t m0 x2b h4 y73 ff1 fs1 fc1 sc0 ls0">1</div><div class="t m0 x2b h4 y74 ff1 fs1 fc1 sc0 ls0">0</div><div class="t m0 x3a h7 y75 ff2 fs1 fc1 sc0 ls0">1</div><div class="t m0 x3a h22 y71 ff2 fs1 fc1 sc0 ls7e">A<span class="ff4 fs4 ls7f v3">+</span><span class="ff3 ls7a v3">c</span><span class="ff1 fs2 lse v4">3</span><span class="ls0 v5">0</span></div><div class="t m0 x3b h7 y71 ff2 fs1 fc1 sc0 ls0">@</div><div class="t m0 x23 h4 y72 ff1 fs1 fc1 sc0 ls0">0</div><div class="t m0 x23 h4 y73 ff1 fs1 fc1 sc0 ls0">0</div><div class="t m0 x23 h4 y74 ff1 fs1 fc1 sc0 ls0">1</div><div class="t m0 x32 h7 y75 ff2 fs1 fc1 sc0 ls0">1</div><div class="t m0 x32 h26 y71 ff2 fs1 fc1 sc0 ls80">A<span class="ff4 fs4 ls79 v3">=</span><span class="ls0 v5">0</span></div><div class="t m0 x3c h7 y76 ff2 fs1 fc1 sc0 ls0">@</div><div class="t m0 x3d h4 y77 ff1 fs1 fc1 sc0 ls0">0</div><div class="t m0 x3d h4 y78 ff1 fs1 fc1 sc0 ls0">0</div><div class="t m0 x3d h4 y79 ff1 fs1 fc1 sc0 ls0">0</div><div class="t m0 x3e h7 y7a ff2 fs1 fc1 sc0 ls0">1</div><div class="t m0 x3e h22 y76 ff2 fs1 fc1 sc0 ls81">A<span class="ff5 fs4 ls82 v3">$</span><span class="ls0 v5">0</span></div><div class="t m0 x28 h7 y76 ff2 fs1 fc1 sc0 ls0">@</div><div class="t m0 x1b h4 y77 ff3 fs1 fc1 sc0 ls83">c<span class="ff1 fs2 ls0 v1">1</span></div><div class="t m0 x1b h4 y6e ff3 fs1 fc1 sc0 ls83">c<span class="ff1 fs2 ls0 v1">2</span></div><div class="t m0 x1b h4 y7b ff3 fs1 fc1 sc0 ls83">c<span class="ff1 fs2 ls0 v1">3</span></div><div class="t m0 x3f h7 y7c ff2 fs1 fc1 sc0 ls0">1</div><div class="t m0 x3f h27 y7d ff2 fs1 fc1 sc0 ls81">A<span class="ff4 fs4 ls14 v3">=</span><span class="ls0 v5">0</span></div><div class="t m0 x40 h7 y76 ff2 fs1 fc1 sc0 ls0">@</div><div class="t m0 x41 h4 y77 ff1 fs1 fc1 sc0 ls0">0</div><div class="t m0 x41 h4 y78 ff1 fs1 fc1 sc0 ls0">0</div><div class="t m0 x41 h4 y79 ff1 fs1 fc1 sc0 ls0">0</div><div class="t m0 x42 h7 y7a ff2 fs1 fc1 sc0 ls0">1</div><div class="t m0 x42 h7 y76 ff2 fs1 fc1 sc0 ls2e">A<span class="ff7 ls0 v3">,</span></div><div class="t m0 x8 h28 y7e ff1 fs1 fc1 sc0 ls0 ws7">o que implica que <span class="ff3 ls83">c</span><span class="fs2 ls84 v1">1</span><span class="ff4 fs4 ls1c v0">=<span class="ff3 fs1 ls83">c</span></span><span class="fs2 ls85 v1">2</span><span class="ff4 fs4 ls1c v0">=<span class="ff3 fs1 ls86">c</span></span><span class="fs2 ls87 v1">3</span><span class="ff4 fs4 ls1c v0">=</span><span class="ws3 v0">0<span class="ff7">.</span></span></div><div class="t m0 x5 h5 y4e ff1 fs2 fc1 sc0 ls0">8</div></div><div class="c x0 ya w2 h2"><div class="t m0 x7 h3 yb ff1 fs0 fc0 sc0 ls0 wsd">In<span class="_0 blank"></span>depen<span class="_0 blank"></span>dên<span class="_0 blank"></span>cia<span class="_6 blank"> </span>Li<span class="_0 blank"></span>nea<span class="_4 blank"></span>r:<span class="_1 blank"> </span>E<span class="_0 blank"></span>xe<span class="_0 blank"></span>mp<span class="_0 blank"></span>los</div><div class="t m0 x8 h4 y7f ff1 fs1 fc2 sc0 ls0 ws7">Exemplo 2:<span class="_3 blank"> </span><span class="fc1 ws8">Os veto<span class="_0 blank"></span>res</span></div><div class="t m0 xd h29 y80 ff6 fs1 fc1 sc0 ls88">w<span class="ff1 fs2 ls89 v1">1</span><span class="ff4 fs4 ls79 v0">=</span><span class="ff2 ls0 v2">0</span></div><div class="t m0 x6 h7 y81 ff2 fs1 fc1 sc0 ls0">@</div><div class="t m0 x43 h4 y82 ff1 fs1 fc1 sc0 ls0">1</div><div class="t m0 x43 h4 y83 ff1 fs1 fc1 sc0 ls0">2</div><div class="t m0 x43 h4 y84 ff1 fs1 fc1 sc0 ls0">3</div><div class="t m0 x44 h7 y85 ff2 fs1 fc1 sc0 ls0">1</div><div class="t m0 x44 h22 y81 ff2 fs1 fc1 sc0 ls2e">A<span class="ff7 ls74 v3">,<span class="ff6 ls8a">w<span class="ff1 fs2 ls8b v1">2</span><span class="ff4 fs4 ls14 v0">=</span></span></span><span class="ls0 v5">0</span></div><div class="t m0 x45 h7 y81 ff2 fs1 fc1 sc0 ls0">@</div><div class="t m0 x12 h4 y82 ff1 fs1 fc1 sc0 ls0">4</div><div class="t m0 x12 h4 y83 ff1 fs1 fc1 sc0 ls0">5</div><div class="t m0 x12 h4 y84 ff1 fs1 fc1 sc0 ls0">6</div><div class="t m0 x46 h7 y85 ff2 fs1 fc1 sc0 ls0">1</div><div class="t m0 x46 h22 y81 ff2 fs1 fc1 sc0 ls2e">A<span class="ff7 ls74 v3">,<span class="ff6 ls88">w<span class="ff1 fs2 ls8c v1">3</span><span class="ff4 fs4 ls79 v0">=</span></span></span><span class="ls0 v5">0</span></div><div class="t m0 x47 h7 y81 ff2 fs1 fc1 sc0 ls0">@</div><div class="t m0 x48 h4 y82 ff1 fs1 fc1 sc0 ls0">7</div><div class="t m0 x48 h4 y83 ff1 fs1 fc1 sc0 ls0">8</div><div class="t m0 x48 h4 y84 ff1 fs1 fc1 sc0 ls0">9</div><div class="t m0 x49 h7 y85 ff2 fs1 fc1 sc0 ls0">1</div><div class="t m0 x49 h7 y81 ff2 fs1 fc1 sc0 ls0">A</div><div class="t m0 x8 h23 y86 ff1 fs1 fc1 sc0 ls0 ws36">em <span class="ff8 ls3e">R</span><span class="fs2 ls38 v7">3</span><span class="ws37">são<span class="_2 blank"> </span>linea<span class="_0 blank"></span>rmente<span class="_2 blank"> </span>dep endentes<span class="_2 blank"> </span>p ois<span class="_2 blank"> </span>a<span class="_2 blank"> </span>equação:</span></div><div class="t m0 xd h24 y87 ff3 fs1 fc1 sc0 ls8d">c<span class="ff1 fs2 ls8e v1">1</span><span class="ff2 ls0 v2">0</span></div><div class="t m0 x1f h7 y88 ff2 fs1 fc1 sc0 ls0">@</div><div class="t m0 x6 h4 y89 ff1 fs1 fc1 sc0 ls0">1</div><div class="t m0 x6 h4 y8a ff1 fs1 fc1 sc0 ls0">2</div><div class="t m0 x6 h4 y8b ff1 fs1 fc1 sc0 ls0">3</div><div class="t m0 x0 h7 y8c ff2 fs1 fc1 sc0 ls0">1</div><div class="t m0 x0 h2a y88 ff2 fs1 fc1 sc0 ls8f">A<span class="ff4 fs4 ls10 v3">+</span><span class="ff3 ls86 v3">c</span><span class="ff1 fs2 ls90 v4">2</span><span class="ls0 v5">0</span></div><div class="t m0 x4a h7 y8d ff2 fs1 fc1 sc0 ls0">@</div><div class="t m0 x4b h4 y8e ff1 fs1 fc1 sc0 ls0">4</div><div class="t m0 x4b h4 y8f ff1 fs1 fc1 sc0 ls0">5</div><div class="t m0 x4b h4 y90 ff1 fs1 fc1 sc0 ls0">6</div><div class="t m0 x4c h7 y91 ff2 fs1 fc1 sc0 ls0">1</div><div class="t m0 x4c h22 y8d ff2 fs1 fc1 sc0 ls91">A<span class="ff4 fs4 ls10 v3">+</span><span class="ff3 ls7a v3">c</span><span class="ff1 fs2 lse v4">3</span><span class="ls0 v5">0</span></div><div class="t m0 x4d h7 y8d ff2 fs1 fc1 sc0 ls0">@</div><div class="t m0 x4e h4 y8e ff1 fs1 fc1 sc0 ls0">7</div><div class="t m0 x4e h4 y8f ff1 fs1 fc1 sc0 ls0">8</div><div class="t m0 x4e h4 y90 ff1 fs1 fc1 sc0 ls0">9</div><div class="t m0 x4f h7 y91 ff2 fs1 fc1 sc0 ls0">1</div><div class="t m0 x4f h2b y8d ff2 fs1 fc1 sc0 ls92">A<span class="ff4 fs4 ls14 v3">=</span><span class="ls0 v5">0</span></div><div class="t m0 x50 h7 y92 ff2 fs1 fc1 sc0 ls0">@</div><div class="t m0 x1b h4 y93 ff1 fs1 fc1 sc0 ls0">0</div><div class="t m0 x1b h4 y94 ff1 fs1 fc1 sc0 ls0">0</div><div class="t m0 x1b h4 y95 ff1 fs1 fc1 sc0 ls0">0</div><div class="t m0 x51 h7 y96 ff2 fs1 fc1 sc0 ls0">1</div><div class="t m0 x51 h7 y92 ff2 fs1 fc1 sc0 ls0">A</div><div class="t m0 x8 h2c y97 ff1 fs1 fc1 sc0 ls0 ws38">é satisfeita pa<span class="_0 blank"></span>ra <span class="ff3 ls7a">c</span><span class="fs2 ls93 v1">1</span><span class="ff4 fs4 ls1c v0">=</span><span class="ws39 v0">1, <span class="ff3 ls83">c</span><span class="fs2 ls94 v1">2</span><span class="ff4 fs4 ls14">=<span class="ff5 ls95">\ue000</span></span><span class="ws7">2 e <span class="ff3 ls86">c</span><span class="fs2 ls96 v1">3</span><span class="ff4 fs4 ls1c">=</span><span class="ws3">1<span class="ff7">.</span></span></span></span></div><div class="t m0 x5 h5 y14 ff1 fs2 fc1 sc0 ls0">9</div></div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div> <div id="pf4" class="pf w0 h0" data-page-no="4"><div class="pc pc4 w0 h0"><img fetchpriority="low" loading="lazy" class="bi x0 y0 w1 h1" alt="" src="https://files.passeidireto.com/e1597c04-9ff2-4637-afbf-78fed828f3b2/bg4.png"><div class="c x0 y1 w2 h2"><div class="t m0 x7 h3 y15 ff1 fs0 fc0 sc0 ls0 ws3a">V<span class="_4 blank"></span>eri<span class="_0 blank"></span>\u2026ca<span class="_0 blank"></span>ndo<span class="_6 blank"> </span>In<span class="_0 blank"></span>depen<span class="_0 blank"></span>dên<span class="_0 blank"></span>cia<span class="_6 blank"> </span>Li<span class="_0 blank"></span>nea<span class="_4 blank"></span>r</div><div class="t m0 x8 h4 y98 ff1 fs1 fc1 sc0 ls0 ws7">Considere os veto<span class="_0 blank"></span>res <span class="ff6 ls8a">w</span><span class="fs2 ls97 v1">1</span><span class="ls98">,<span class="ff6 ls99">w</span><span class="fs2 ls38 v1">2</span><span class="ls9a">e<span class="ff6 ls99">w</span><span class="fs2 ls37 v1">3</span></span></span><span class="ws2">do exemplo anterior.<span class="_3 blank"> </span>De fo<span class="_0 blank"></span>rma</span></div><div class="t m0 x8 h4 y99 ff1 fs1 fc1 sc0 ls0 ws2">geral, pa<span class="_0 blank"></span>ra determinarmos se esses veto<span class="_0 blank"></span>res são linearmente</div><div class="t m0 x8 h4 y9a ff1 fs1 fc1 sc0 ls0 ws12">dep endentes<span class="_2 blank"> </span>ou<span class="_2 blank"> </span>indep endentes,<span class="_2 blank"> </span>p<span class="_0 blank"></span>ro cedemos<span class="_2 blank"> </span>da<span class="_2 blank"> </span>seguinte<span class="_2 blank"> </span>maneira...</div><div class="t m0 x8 h4 y9b ff1 fs1 fc1 sc0 ls0 ws7">Primeiro, seguindo a de\u2026nição anterio<span class="_0 blank"></span>r, escreva:</div><div class="t m0 xd h2d y9c ff3 fs1 fc1 sc0 ls7a">c<span class="ff1 fs2 lse v1">1</span><span class="ff2 ls0 v2">0</span></div><div class="t m0 x1f h7 y9d ff2 fs1 fc1 sc0 ls0">@</div><div class="t m0 x6 h4 y9e ff1 fs1 fc1 sc0 ls0">1</div><div class="t m0 x6 h4 y9f ff1 fs1 fc1 sc0 ls0">2</div><div class="t m0 x6 h4 ya0 ff1 fs1 fc1 sc0 ls0">3</div><div class="t m0 x0 h7 ya1 ff2 fs1 fc1 sc0 ls0">1</div><div class="t m0 x0 h2e y9d ff2 fs1 fc1 sc0 ls9b">A<span class="ff4 fs4 ls10 v3">+</span><span class="ff3 ls86 v3">c</span><span class="ff1 fs2 ls90 v4">2</span><span class="ls0 v5">0</span></div><div class="t m0 x4a h7 ya2 ff2 fs1 fc1 sc0 ls0">@</div><div class="t m0 x4b h4 ya3 ff1 fs1 fc1 sc0 ls0">4</div><div class="t m0 x4b h4 ya4 ff1 fs1 fc1 sc0 ls0">5</div><div class="t m0 x4b h4 ya5 ff1 fs1 fc1 sc0 ls0">6</div><div class="t m0 x4c h7 ya1 ff2 fs1 fc1 sc0 ls0">1</div><div class="t m0 x4c h2e y9d ff2 fs1 fc1 sc0 ls91">A<span class="ff4 fs4 ls10 v3">+</span><span class="ff3 ls7a v3">c</span><span class="ff1 fs2 lse v4">3</span><span class="ls0 v5">0</span></div><div class="t m0 x4d h7 ya2 ff2 fs1 fc1 sc0 ls0">@</div><div class="t m0 x4e h4 ya3 ff1 fs1 fc1 sc0 ls0">7</div><div class="t m0 x4e h4 ya4 ff1 fs1 fc1 sc0 ls0">8</div><div class="t m0 x4e h4 ya5 ff1 fs1 fc1 sc0 ls0">9</div><div class="t m0 x4f h7 ya1 ff2 fs1 fc1 sc0 ls0">1</div><div class="t m0 x4f h2f y9d ff2 fs1 fc1 sc0 ls92">A<span class="ff4 fs4 ls79 v3">=</span><span class="ls0 v5">0</span></div><div class="t m0 x50 h7 ya6 ff2 fs1 fc1 sc0 ls0">@</div><div class="t m0 x1b h4 ya7 ff1 fs1 fc1 sc0 ls0">0</div><div class="t m0 x1b h4 ya8 ff1 fs1 fc1 sc0 ls0">0</div><div class="t m0 x1b h4 ya9 ff1 fs1 fc1 sc0 ls0">0</div><div class="t m0 x51 h7 yaa ff2 fs1 fc1 sc0 ls0">1</div><div class="t m0 x51 h7 ya6 ff2 fs1 fc1 sc0 ls0">A</div><div class="t m0 x8 h4 yab ff1 fs1 fc1 sc0 ls0 ws8">e note que p<span class="_5 blank"> </span>odemos re-expressa<span class="_4 blank"></span>r es<span class="_5 blank"> </span>ta equação como:</div><div class="t m0 x52 h7 yac ff2 fs1 fc1 sc0 ls0">8</div><div class="t m0 x52 h7 yad ff2 fs1 fc1 sc0 ls0"><</div><div class="t m0 x52 h7 yae ff2 fs1 fc1 sc0 ls0">:</div><div class="t m0 x2d h30 yaf ff1 fs1 fc1 sc0 ls0 ws3">1<span class="ff3 ls83">c</span><span class="fs2 ls9c v1">1</span><span class="ff4 fs4 ls10 v0">+</span><span class="v0">4<span class="ff3 ls9d">c</span><span class="fs2 ls9e v1">2</span><span class="ff4 fs4 ls10">+</span>7<span class="ff3 ls83">c</span><span class="fs2 ls9f v1">3</span><span class="ff4 fs4 ls1c">=</span>0</span></div><div class="t m0 x2d h31 yb0 ff1 fs1 fc1 sc0 ls0 ws3">2<span class="ff3 ls83">c</span><span class="fs2 lsa0 v1">1</span><span class="ff4 fs4 ls10 v0">+</span><span class="v0">5<span class="ff3 ls9d">c</span><span class="fs2 ls9e v1">2</span><span class="ff4 fs4 ls10">+</span>8<span class="ff3 ls83">c</span><span class="fs2 ls9f v1">3</span><span class="ff4 fs4 ls1c">=</span>0</span></div><div class="t m0 x2d h9 yb1 ff1 fs1 fc1 sc0 ls0 ws3">3<span class="ff3 ls83">c</span><span class="fs2 lsa0 v1">1</span><span class="ff4 fs4 ls10 v0">+</span><span class="v0">6<span class="ff3 ls9d">c</span><span class="fs2 ls9e v1">2</span><span class="ff4 fs4 ls10">+</span>9<span class="ff3 ls83">c</span><span class="fs2 ls9f v1">3</span><span class="ff4 fs4 ls1c">=</span>0</span></div><div class="t m0 x22 h32 yb2 ff5 fs4 fc1 sc0 lsa1">$<span class="ff2 fs1 ls0 v2">0</span></div><div class="t m0 x32 h7 yb3 ff2 fs1 fc1 sc0 ls0">@</div><div class="t m0 x33 h4 yb4 ff1 fs1 fc1 sc0 lsa2 ws3b">147<span class="_8 blank"></span></div><div class="t m0 x33 h4 yb5 ff1 fs1 fc1 sc0 lsa2 ws3b">258<span class="_8 blank"></span></div><div class="t m0 x33 h4 yb6 ff1 fs1 fc1 sc0 lsa2 ws3b">369<span class="_8 blank"></span></div><div class="t m0 x53 h7 yb7 ff2 fs1 fc1 sc0 ls0">1</div><div class="t m0 x53 hc yb8 ff2 fs1 fc1 sc0 lsa3">A<span class="ls0 v5">0</span></div><div class="t m0 x54 h7 yb8 ff2 fs1 fc1 sc0 ls0">@</div><div class="t m0 x16 h4 yb9 ff3 fs1 fc1 sc0 ls9d">c<span class="ff1 fs2 ls0 v1">1</span></div><div class="t m0 x16 h4 yba ff3 fs1 fc1 sc0 ls9d">c<span class="ff1 fs2 ls0 v1">2</span></div><div class="t m0 x16 h4 ybb ff3 fs1 fc1 sc0 ls9d">c<span class="ff1 fs2 ls0 v1">3</span></div><div class="t m0 x55 h7 ybc ff2 fs1 fc1 sc0 ls0">1</div><div class="t m0 x55 h33 ybd ff2 fs1 fc1 sc0 lsa4">A<span class="ff4 fs4 ls79 v3">=</span><span class="ls0 v5">0</span></div><div class="t m0 x56 h7 yb3 ff2 fs1 fc1 sc0 ls0">@</div><div class="t m0 x57 h4 yb4 ff1 fs1 fc1 sc0 ls0">0</div><div class="t m0 x57 h4 yb5 ff1 fs1 fc1 sc0 ls0">0</div><div class="t m0 x57 h4 yb6 ff1 fs1 fc1 sc0 ls0">0</div><div class="t m0 x58 h7 yb7 ff2 fs1 fc1 sc0 ls0">1</div><div class="t m0 x58 h7 yb8 ff2 fs1 fc1 sc0 ls0">A</div><div class="t m0 x8 he ybe ff1 fs1 fc1 sc0 ls0 ws3c">Como<span class="_2 blank"> </span>p o demos<span class="_2 blank"> </span>veri\u2026ca<span class="_0 blank"></span>r<span class="_2 blank"> </span>se<span class="_2 blank"> </span><span class="ff3 ls86">c</span><span class="fs2 ls78 v1">1</span><span class="ff4 fs4 ls1c">=</span><span class="ff3 ls86">c</span><span class="fs2 ls87 v1">2</span><span class="ff4 fs4 ls1c">=</span><span class="ff3 ls8d">c</span><span class="fs2 lsa5 v1">3</span><span class="ff4 fs4 ls1c">=</span><span class="ws7">0 é a <span class="fc2 ws3d">única </span><span class="ws2">solução deste</span></span></div><div class="t m0 x8 h4 ybf ff1 fs1 fc1 sc0 ls0 ws3">sistema?</div><div class="t m0 x59 h5 y32 ff1 fs2 fc1 sc0 ls0 ws3e">10</div></div><div class="c x0 y7 w2 h2"><div class="t m0 x7 h3 y33 ff1 fs0 fc0 sc0 ls0 ws3f">V<span class="_4 blank"></span>eri<span class="_0 blank"></span>\u2026ca<span class="_0 blank"></span>ndo<span class="_6 blank"> </span>In<span class="_0 blank"></span>depen<span class="_0 blank"></span>dên<span class="_0 blank"></span>cia<span class="_6 blank"> </span>Li<span class="_0 blank"></span>nea<span class="_4 blank"></span>r</div><div class="t m0 x8 h4 yc0 ff1 fs1 fc1 sc0 ls0 ws3">(Cont.)</div><div class="t m0 x36 h34 yc1 ffa fs7 fc0 sc0 lsa6">I<span class="ff1 fs5 fc1 ls0 ws28 v1">Note que o sistema acima é um <span class="fc2 ws40">sistema homogêneo<span class="fc1 ls54">,<span class="ff3 ls0 ws1f">A<span class="ff6 lsa7">x<span class="ff4 fs6 ls6d">=</span><span class="ls0">0<span class="ff1 ws41">. Neste</span></span></span></span></span></span></span></div><div class="t m0 x2d h1e yc2 ff1 fs5 fc1 sc0 ls0 ws42">caso,<span class="_7 blank"> </span>sab emos<span class="_7 blank"> </span>que:</div><div class="t m0 x5a h35 yc3 ff3 fs8 fc0 sc0 lsa8">i<span class="ffb lsa9">.<span class="ff1 fc1 ls0 ws43">Se<span class="_7 blank"> </span>p<span class="_5 blank"> </span>osto<span class="_7 blank"> </span><span class="ff3 lsaa">A<span class="ff4 fs9 lsab">=</span><span class="lsac">n<span class="fsa lsad vf">o</span></span></span><span class="ws44">inc ó g n i ta s ,<span class="_7 blank"> </span>ent ã o<span class="_7 blank"> </span>o<span class="_7 blank"> </span>sist e m a<span class="_7 blank"> </span>p<span class="_5 blank"> </span>oss u i<span class="_7 blank"> </span>um a<span class="_7 blank"> </span><span class="fc2">ún ic a<span class="_7 blank"> </span>so lu ç ã o </span>,</span></span></span></div><div class="t m0 x5b h36 yc4 ff6 fs8 fc1 sc0 lsae">x<span class="ff4 fs9 lsab v0">=</span><span class="ls0 ws45 v0">0<span class="ff1">;</span></span></div><div class="t m0 x5c h37 yc5 ff3 fs8 fc0 sc0 ls0 ws46">ii <span class="ffb lsaf">.</span><span class="ff1 fc1 ws47">Se<span class="_7 blank"> </span>p<span class="_5 blank"> </span>osto<span class="_7 blank"> </span><span class="ff3 lsb0">A<span class="ffc fs9 lsb1"><</span><span class="lsb2">n<span class="fsa lsb3 vf">o</span></span></span><span class="ws48">in c ó g<span class="_5 blank"> </span>nitas ,<span class="_7 blank"> </span>en t ã o<span class="_7 blank"> </span>o<span class="_7 blank"> </span>sist e<span class="_5 blank"> </span>ma<span class="_7 blank"> </span>p o s s u i<span class="_7 blank"> </span><span class="fc2 ws44">in\u2026<span class="_5 blank"> </span>nita s<span class="_7 blank"> </span>sol u çõ e s </span>.</span></span></div><div class="t m0 x36 h38 yc6 ffa fs7 fc0 sc0 lsa6">I<span class="ff1 fs5 fc1 ls0 ws28 v1">Assim, reduzindo a matriz de co<span class="_5 blank"> </span>e\u2026cientes <span class="ff3 lsb4">A</span><span class="ws49">à su<span class="_0 blank"></span>a forma escalonada</span></span></div><div class="t m0 x2d h1e yc7 ff1 fs5 fc1 sc0 ls0 ws26">p o<span class="_0 blank"></span>r<span class="_7 blank"> </span>linhas,<span class="_7 blank"> </span>obtemos:</div><div class="t m0 x5d h39 yc8 ff2 fs5 fc1 sc0 ls0">0</div><div class="t m0 x5d h39 yc9 ff2 fs5 fc1 sc0 ls0">@</div><div class="t m0 x3 h1e yca ff1 fs5 fc1 sc0 ls0 ws4a">1 4 7</div><div class="t m0 x3 h1e ycb ff1 fs5 fc1 sc0 ls0 ws4a">2 5 8</div><div class="t m0 x3 h1e ycc ff1 fs5 fc1 sc0 ls0 ws4a">3 6 9</div><div class="t m0 x5e h39 ycd ff2 fs5 fc1 sc0 ls0">1</div><div class="t m0 x5e h3a yce ff2 fs5 fc1 sc0 lsb5">A<span class="ff4 fsb lsb6 v10">(<span class="ff9 fs2 ls0 ws4b">...<span class="ff4 fsb">)</span></span></span></div><div class="t m0 x45 h3b ycf ff5 fs6 fc1 sc0 lsb7">)<span class="ff2 fs5 ls0 v11">0</span></div><div class="t m0 x5f h39 yd0 ff2 fs5 fc1 sc0 ls0">@</div><div class="t m0 x26 h1e yd1 ff1 fs5 fc1 sc0 ls0 ws4c">1 4<span class="_9 blank"> </span>7</div><div class="t m0 x26 h1e yd2 ff1 fs5 fc1 sc0 lsb8">0<span class="ff5 fs6 lsb9">\ue000</span>3<span class="ff5 fs6 lsba">\ue000</span><span class="ls0">6</span></div><div class="t m0 x26 h1e yd3 ff1 fs5 fc1 sc0 ls0 ws4c">0 0<span class="_9 blank"> </span>0</div><div class="t m0 x1b h39 yd4 ff2 fs5 fc1 sc0 ls0">1</div><div class="t m0 x1b h39 yd5 ff2 fs5 fc1 sc0 ls0">A</div><div class="t m0 x2d h20 yd6 ff1 fs5 fc1 sc0 ls0 ws42">P<span class="_0 blank"></span>ortanto,<span class="_7 blank"> </span>posto<span class="_7 blank"> </span><span class="ff3 lsbb">A<span class="ff4 fs6 ls67">=</span></span><span class="lsbc">2<span class="ffc fs6 ls6b"><</span></span><span class="ws1f">3<span class="ff7">.</span></span></div><div class="t m0 x59 h5 yd7 ff1 fs2 fc1 sc0 ls0 ws3e">11</div></div><div class="c x0 ya w2 h2"><div class="t m0 x7 h3 yb ff1 fs0 fc0 sc0 ls0 ws3f">V<span class="_4 blank"></span>eri<span class="_0 blank"></span>\u2026ca<span class="_0 blank"></span>ndo<span class="_6 blank"> </span>In<span class="_0 blank"></span>depen<span class="_0 blank"></span>dên<span class="_0 blank"></span>cia<span class="_6 blank"> </span>Li<span class="_0 blank"></span>nea<span class="_4 blank"></span>r</div><div class="t m0 x8 h4 yd8 ff1 fs1 fc1 sc0 ls0 ws3">(Cont.)</div><div class="t m0 x36 h3c yd9 ffa fs7 fc0 sc0 lsa6">I<span class="ff1 fs5 fc1 ls0 ws28 v1">Desta fo<span class="_0 blank"></span>rma, o sistema p<span class="_5 blank"> </span>ossui uma in\u2026nidade de soluções não-nulas.</span></div><div class="t m0 x2d h1e yda ff1 fs5 fc1 sc0 ls0 ws23">P<span class="_0 blank"></span>or exemplo, é possível veri\u2026car que uma tal solução po<span class="_5 blank"> </span>deria ser a</div><div class="t m0 x2d h20 ydb ff1 fs5 fc1 sc0 ls0 ws4d">seguinte: <span class="ff3 ls6f">c</span><span class="fs2 ls70 v1">1</span><span class="ff4 fs6 ls6d">=</span><span class="ws4e">1, <span class="ff3 ls61">c</span><span class="fs2 ls70 v1">2</span><span class="ff4 fs6 lsbd">=<span class="ff5 lsb9">\ue000</span></span><span class="ws28">2 e <span class="ff3 ls5e">c</span><span class="fs2 ls70 v1">3</span><span class="ff4 fs6 ls67">=</span><span class="ws1f">1<span class="ff7">.</span></span></span></span></div><div class="t m0 x36 h3c ydc ffa fs7 fc0 sc0 lsa6">I<span class="ff1 fs5 fc1 ls0 ws1e v1">P<span class="_0 blank"></span>ortanto, concluímos que os veto<span class="_0 blank"></span>res <span class="ff6 lsbe">w<span class="ff1 fs2 ls16 v1">1</span><span class="ff1 ls54">,</span><span class="lsbf">w<span class="ff1 fs2 lsc0 v1">2</span><span class="ff1 lsc1">e</span><span class="lsc2">w<span class="ff1 fs2 lsc3 v1">3</span></span></span></span><span class="ws28">são linearmente</span></span></div><div class="t m0 x2d h1e ydd ff1 fs5 fc1 sc0 ls0 ws26">dep endentes.</div><div class="t m0 x59 h5 yde ff1 fs2 fc1 sc0 ls0 ws3e">12</div></div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div> <div id="pf5" class="pf w0 h0" data-page-no="5"><div class="pc pc5 w0 h0"><img fetchpriority="low" loading="lazy" class="bi x0 y0 w1 h1" alt="" src="https://files.passeidireto.com/e1597c04-9ff2-4637-afbf-78fed828f3b2/bg5.png"><div class="c x0 y1 w2 h2"><div class="t m0 x7 h3 y15 ff1 fs0 fc0 sc0 ls0 ws3a">V<span class="_4 blank"></span>eri<span class="_0 blank"></span>\u2026ca<span class="_0 blank"></span>ndo<span class="_6 blank"> </span>In<span class="_0 blank"></span>depen<span class="_0 blank"></span>dên<span class="_0 blank"></span>cia<span class="_6 blank"> </span>Li<span class="_0 blank"></span>nea<span class="_4 blank"></span>r</div><div class="t m0 x8 h3d ydf ff1 fs1 fc2 sc0 ls0 ws4f">T<span class="_4 blank"></span>eo<span class="_0 blank"></span>rema: <span class="fc1 ws8">Os vetores <span class="ff6 ws3">v</span><span class="fs2 ls16 v1">1</span><span class="ff7 ls3b">,</span><span class="ff6 ws3">v</span><span class="fs2 ls16 v1">2</span><span class="ff7 ws18">, ...,<span class="_7 blank"> </span><span class="ff6 ws3">v<span class="ff3 fs2 lsc4 v1">k</span></span></span><span class="ws36">em <span class="ff8 ls45">R<span class="ff3 fs2 lsc5 v7">n</span></span><span class="ws7">são linearmente</span></span></span></div><div class="t m0 x8 h4 ye0 ff1 fs1 fc1 sc0 ls0 ws2">dep<span class="_5 blank"> </span>endentes se, e somente se, o sistema linea<span class="_0 blank"></span>r homogêneo:</div><div class="t m0 x60 h3e ye1 ff3 fs1 fc1 sc0 lsc6">A<span class="ff2 ls0 v12">0</span></div><div class="t m0 xb h7 ye2 ff2 fs1 fc1 sc0 ls0">B</div><div class="t m0 xb h7 ye3 ff2 fs1 fc1 sc0 ls0">@</div><div class="t m0 x4c h4 ye4 ff3 fs1 fc1 sc0 ls86">c<span class="ff1 fs2 ls0 v1">1</span></div><div class="t m0 x61 h4 ye5 ff1 fs1 fc1 sc0 ls0">.</div><div class="t m0 x61 h4 ye6 ff1 fs1 fc1 sc0 ls0">.</div><div class="t m0 x61 h4 ye7 ff1 fs1 fc1 sc0 ls0">.</div><div class="t m0 x4c h4 ye8 ff3 fs1 fc1 sc0 ls86">c<span class="fs2 ls0 v1">k</span></div><div class="t m0 x62 h7 ye9 ff2 fs1 fc1 sc0 ls0">1</div><div class="t m0 x62 h7 yea ff2 fs1 fc1 sc0 ls0">C</div><div class="t m0 x62 h3f yeb ff2 fs1 fc1 sc0 lsc7">A<span class="ff4 fs4 lsc8 v13">=</span><span class="ff6 ls0 v13">0</span></div><div class="t m0 x8 h40 yec ff1 fs1 fc1 sc0 ls0 ws50">tem uma solução não-nula <span class="ff4 fs4 lsc9 v0">(</span><span class="ff3 ls86 v0">c</span><span class="fs2 ls5c v1">1</span><span class="ff7 ws51 v0">, ..., <span class="ff3 ls86">c<span class="fs2 lsca v1">k</span><span class="ff4 fs4 lscb v0">)</span></span><span class="lscc">,</span><span class="ff1 ws52">onde <span class="ff3 lscd">A</span><span class="ws53">é a matriz <span class="ff3 lsce">n<span class="ff5 fs4 lscf">\ue002</span><span class="lsd0">k</span></span><span class="ws54">, cujas</span></span></span></span></div><div class="t m0 x8 h4 yed ff1 fs1 fc1 sc0 ls0 ws8">colunas são os veto<span class="_0 blank"></span>res <span class="ff6 ws3">v</span><span class="fs2 ls16 v1">1</span><span class="ff7 ls3b">,</span><span class="ff6 ws3">v</span><span class="fs2 ls18 v1">2</span><span class="ff7 ws18">, ...,<span class="_2 blank"> </span><span class="ff6 ws3">v<span class="ff3 fs2 lsd1 v1">k</span><span class="ffd">:</span></span></span></div><div class="t m0 x3 h41 yee ff3 fs1 fc1 sc0 lsd2">A<span class="ff4 fs4 ls79 v0">=</span><span class="ff2 lsd3 v14">\ue000</span><span class="ff6 ls0 ws3 v0">v<span class="ff1 fs2 lsd4 v1">1</span>v<span class="ff1 fs2 lsd5 v1">2</span><span class="ff5 fs4 ws55 v0">\ue001 \ue001 \ue001<span class="_a blank"> </span></span><span class="v0">v</span></span><span class="fs2 lsd6 v1">k</span><span class="ff2 ls0 v14">\ue001</span></div><div class="t m0 x8 h23 yef ff1 fs1 fc1 sc0 ls0 ws13">Assim, temos que os veto<span class="_0 blank"></span>res <span class="ff6 ws3">v</span><span class="fs2 ls18 v1">1</span><span class="ff7 ls3b">,</span><span class="ff6 ws3">v</span><span class="fs2 ls16 v1">2</span><span class="ff7 ws51">, ...,<span class="_2 blank"> </span><span class="ff6 ws3">v<span class="ff3 fs2 lsd7 v1">k</span></span></span><span class="ws56">em <span class="ff8 ls4b">R<span class="ff3 fs2 lsd8 v7">n</span></span><span class="ws7">são linearmente</span></span></div><div class="t m0 x8 h4 yf0 ff1 fs1 fc1 sc0 ls0 ws12">indep endentes<span class="_2 blank"> </span>se,<span class="_2 blank"> </span>e<span class="_2 blank"> </span>somente<span class="_2 blank"> </span>se:</div><div class="t m0 x43 h42 yf1 ff1 fs1 fc1 sc0 ls0 ws57">p osto <span class="ff4 fs4 lsd9 v0">(</span><span class="ff3 lsda v0">A<span class="ff4 fs4 lsdb v0">)<span class="lsdc v0">=</span></span><span class="lsdd">k</span></span><span class="ws7 v0">(= n<span class="ff3 fs2 lsde v7">o</span><span class="ws58">vetores)<span class="_b blank"> </span>(<span class="ff5 fs4 lsdf">\ue003</span>)</span></span></div><div class="t m0 x59 h5 y32 ff1 fs2 fc1 sc0 ls0 ws3e">13</div></div><div class="c x0 y7 w2 h2"><div class="t m0 x7 h3 y33 ff1 fs0 fc0 sc0 ls0 ws3f">V<span class="_4 blank"></span>eri<span class="_0 blank"></span>\u2026ca<span class="_0 blank"></span>ndo<span class="_6 blank"> </span>In<span class="_0 blank"></span>depen<span class="_0 blank"></span>dên<span class="_0 blank"></span>cia<span class="_6 blank"> </span>Li<span class="_0 blank"></span>nea<span class="_4 blank"></span>r</div><div class="t m0 x8 he yf2 ff1 fs1 fc1 sc0 ls0 ws7">No caso em que <span class="ff3 lse0">k<span class="ff4 fs4 ls1c">=</span><span class="lse1">n</span></span><span class="ws2">, i.e.<span class="_3 blank"> </span>o número de veto<span class="_0 blank"></span>res é igual à dimensão</span></div><div class="t m0 x8 h23 yf3 ff1 fs1 fc1 sc0 ls0 ws59">do espaço <span class="ff8 ls45">R<span class="ff3 fs2 lse2 v7">n</span></span><span class="ws8">, p<span class="_5 blank"> </span>odemos utilizar o fato de que uma matriz quadrada é</span></div><div class="t m0 x8 h4 yf4 ff1 fs1 fc2 sc0 ls0 ws5a">não-singula<span class="_0 blank"></span>r <span class="fc1 ws8">se, e somente se, o seu </span><span class="ws5b">determinante <span class="fc1 lse3">é</span><span class="ws3">não-nulo<span class="fc1">.</span></span></span></div><div class="t m0 x8 h23 yf5 ff1 fs1 fc2 sc0 ls0 ws5c">T<span class="_4 blank"></span>eo<span class="_0 blank"></span>rema: <span class="fc1 ws2">Um conjunto de <span class="ff3 lse4">n</span><span class="ws5d">veto<span class="_0 blank"></span>res <span class="ff6 ws3">v</span><span class="fs2 ls16 v1">1</span><span class="ff7 ls21">,</span><span class="ff6 ws3">v</span><span class="fs2 ls16 v1">2</span><span class="ff7 ws51">, ...,<span class="_2 blank"> </span><span class="ff6 ws3">v<span class="ff3 fs2 ls3f v1">n</span></span></span><span class="ws5e">em <span class="ff8 lse5">R<span class="ff3 fs2 lse6 v7">n</span></span>é</span></span></span></div><div class="t m0 x8 h4 yf6 ff1 fs1 fc1 sc0 ls0 ws2">linea<span class="_0 blank"></span>rmente indep<span class="_5 blank"> </span>endentes se, e somente se,</div><div class="t m0 x31 h43 yf7 ff1 fs1 fc1 sc0 ls0 ws5f">det <span class="ff2 lse7 v14">\ue000</span><span class="ff6 ws3 v0">v</span><span class="fs2 lse8 v1">1</span><span class="ff6 ws3 v0">v</span><span class="fs2 lse9 v1">2</span><span class="ff5 fs4 ws55 v0">\ue001 \ue001 \ue001<span class="_a blank"> </span><span class="ff6 fs1 ws3">v<span class="ff3 fs2 lsea v1">n</span><span class="ff2 lseb v14">\ue001</span></span><span class="lsec v0">6<span class="ff4 lsed">=</span></span></span><span class="ws58">0<span class="_c blank"> </span>(<span class="ff5 fs4 ws60">\ue003<span class="_5 blank"> </span>\ue003</span>)</span></div><div class="t m0 x59 h5 y9 ff1 fs2 fc1 sc0 ls0 ws3e">14</div></div><div class="c x0 ya w2 h2"><div class="t m0 x7 h3 yb ff1 fs0 fc0 sc0 ls0 ws3f">V<span class="_4 blank"></span>eri<span class="_0 blank"></span>\u2026ca<span class="_0 blank"></span>ndo<span class="_6 blank"> </span>In<span class="_0 blank"></span>depen<span class="_0 blank"></span>dên<span class="_0 blank"></span>cia<span class="_6 blank"> </span>Li<span class="_0 blank"></span>nea<span class="_4 blank"></span>r:<span class="_1 blank"> </span>E<span class="_0 blank"></span>xe<span class="_0 blank"></span>mp<span class="_0 blank"></span>los</div><div class="t m0 x8 h23 yf8 ff1 fs1 fc2 sc0 ls0 ws7">Exemplo 1:<span class="_3 blank"> </span><span class="fc1 ws61">V<span class="_0 blank"></span>etores <span class="ff6 ws3">v</span><span class="fs2 ls9f v1">1</span><span class="ff4 fs4 lsee">=<span class="lsef v0">(</span></span><span class="ws3">1<span class="ff7 lsf0">,</span><span class="lsf1">2<span class="ff4 fs4 lsf2 v0">)</span><span class="ls39">e</span></span><span class="ff6">v</span><span class="fs2 lsf3 v1">2</span><span class="ff4 fs4 lsee">=<span class="lsef v0">(</span></span>3<span class="ff7 lsf4">,</span><span class="lsf1">6<span class="ff4 fs4 lsf2 v0">)</span></span><span class="ws36">em <span class="ff8 lse5">R</span><span class="fs2 ls5c v7">2</span><span class="ff7">.</span></span></span></span></div><div class="t m0 x36 h3c yf9 ffa fs7 fc0 sc0 lsa6">I<span class="ff1 fs5 fc1 ls0 ws28 v1">Note que neste caso o número de veto<span class="_0 blank"></span>res é igual à dimensão do espaço</span></div><div class="t m0 x2d h1e yfa ff1 fs5 fc1 sc0 ls0 ws28">em que eles são rep<span class="_0 blank"></span>resentados.<span class="_3 blank"> </span>Aplicando o teo<span class="_0 blank"></span>rema anterior, escreva:</div><div class="t m0 x37 h44 yfb ff3 fs5 fc1 sc0 lsf5">c<span class="ff1 fs2 lsf6 v1">1</span><span class="ff2 lsf7 v15">\ue012</span><span class="ff1 ls0 vd">1</span></div><div class="t m0 x1d h45 yfc ff1 fs5 fc1 sc0 lsf8">2<span class="ff2 lsf9 v11">\ue013</span><span class="ff4 fs6 ls63 vd">+</span><span class="ff3 ls6f vd">c</span><span class="fs2 lsfa v6">2</span><span class="ff2 lsfb v11">\ue012</span><span class="ls0 v16">3</span></div><div class="t m0 x63 h45 yfd ff1 fs5 fc1 sc0 lsf8">6<span class="ff2 lsfc v11">\ue013</span><span class="ff4 fs6 lsbd vd">=</span><span class="ff2 lsf7 v11">\ue012</span><span class="ls0 v16">0</span></div><div class="t m0 x64 h45 yfe ff1 fs5 fc1 sc0 lsfd">0<span class="ff2 lsfe v11">\ue013</span><span class="ff5 fs6 lsff vd">!</span><span class="ff2 lsf7 v11">\ue012</span><span class="ls0 ws62 v16">1 3</span></div><div class="t m0 x65 h45 yff ff1 fs5 fc1 sc0 ls0 ws63">2<span class="_d blank"> </span>6 <span class="ff2 ws64 v11">\ue013 \ue012<span class="_e blank"> </span></span><span class="ff3 ls5e v16">c</span><span class="fs2 v17">1</span></div><div class="t m0 x48 h46 y100 ff3 fs5 fc1 sc0 ls5e">c<span class="ff1 fs2 ls100 v1">2</span><span class="ff2 ls101 v11">\ue013</span><span class="ff4 fs6 ls102 vd">=</span><span class="ff2 lsfb v11">\ue012</span><span class="ff1 ls0 v16">0</span></div><div class="t m0 x66 h45 y101 ff1 fs5 fc1 sc0 ls103">0<span class="ff2 ls0 v11">\ue013</span></div><div class="t m0 x36 h47 y102 ffa fs7 fc0 sc0 lsa6">I<span class="ff1 fs5 fc1 ls0 ws23 v1">O determinante da matriz de co<span class="_5 blank"> </span>e\u2026cientes é:</span></div><div class="t m0 x67 h39 y103 ff2 fs5 fc1 sc0 ls0">\ue00c</div><div class="t m0 x67 h39 y104 ff2 fs5 fc1 sc0 ls0">\ue00c</div><div class="t m0 x67 h39 y105 ff2 fs5 fc1 sc0 ls0">\ue00c</div><div class="t m0 x67 h39 y106 ff2 fs5 fc1 sc0 ls0">\ue00c</div><div class="t m0 x60 h1e y107 ff1 fs5 fc1 sc0 ls0 ws4a">1 3</div><div class="t m0 x60 h48 y108 ff1 fs5 fc1 sc0 ls0 ws65">2<span class="_d blank"> </span>6 <span class="ff2 v18">\ue00c</span></div><div class="t m0 x4c h39 y109 ff2 fs5 fc1 sc0 ls0">\ue00c</div><div class="t m0 x4c h39 y10a ff2 fs5 fc1 sc0 ls0">\ue00c</div><div class="t m0 x4c h49 y10b ff2 fs5 fc1 sc0 ls104">\ue00c<span class="ff4 fs6 ls6d vf">=</span><span class="ff1 ls0 ws1f vf">1<span class="ff7">.<span class="ff1 ls105">6<span class="ff5 fs6 ls106">\ue000</span></span></span>2<span class="ff7">.</span><span class="ls107">3<span class="ff4 fs6 ls67">=</span></span>0<span class="ff7">,</span></span></div><div class="t m0 x2d h4a y10c ff1 fs5 fc1 sc0 ls0 ws66">i.e.<span class="_6 blank"> </span>o sistema admite uma <span class="fc2 ws28 v0">solução não-nula<span class="fc1 ws23">.<span class="_3 blank"> </span>P<span class="_4 blank"></span>ortanto, os veto<span class="_0 blank"></span>res são</span></span></div><div class="t m0 x2d h1e y10d ff1 fs5 fc2 sc0 ls0 ws26">linea<span class="_0 blank"></span>rmente<span class="_7 blank"> </span>dep endentes<span class="fc1">.</span></div><div class="t m0 x59 h5 y59 ff1 fs2 fc1 sc0 ls0 ws3e">15</div></div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div> <div id="pf6" class="pf w0 h0" data-page-no="6"><div class="pc pc6 w0 h0"><img fetchpriority="low" loading="lazy" class="bi x0 y0 w1 h1" alt="" src="https://files.passeidireto.com/e1597c04-9ff2-4637-afbf-78fed828f3b2/bg6.png"><div class="c x0 y1 w2 h2"><div class="t m0 x7 h3 y15 ff1 fs0 fc0 sc0 ls0 ws3a">V<span class="_4 blank"></span>eri<span class="_0 blank"></span>\u2026ca<span class="_0 blank"></span>ndo<span class="_6 blank"> </span>In<span class="_0 blank"></span>depen<span class="_0 blank"></span>dên<span class="_0 blank"></span>cia<span class="_6 blank"> </span>Li<span class="_0 blank"></span>nea<span class="_4 blank"></span>r:<span class="_1 blank"> </span>E<span class="_0 blank"></span>xe<span class="_0 blank"></span>mp<span class="_0 blank"></span>los</div><div class="t m0 x8 h23 y10e ff1 fs1 fc2 sc0 ls0 ws7">Exemplo 2:<span class="_3 blank"> </span><span class="fc1 ws67">V<span class="_0 blank"></span>etores <span class="ff6 ws3">v</span><span class="fs2 ls9f v1">1</span><span class="ff4 fs4 lsee">=<span class="lsef v0">(</span></span><span class="ws3">2<span class="ff7 lsf0">,</span><span class="lsf1">7<span class="ff4 fs4 lsf2 v0">)</span><span class="ls39">e</span></span><span class="ff6">v</span><span class="fs2 lsf3 v1">2</span><span class="ff4 fs4 lsee">=<span class="lsef v0">(</span></span>1<span class="ff7 lsf4">,</span><span class="lsf1">8<span class="ff4 fs4 lsf2 v0">)</span></span><span class="ws36">em <span class="ff8 lse5">R</span><span class="fs2 ls5c v7">2</span><span class="ff7">.</span></span></span></span></div><div class="t m0 x36 h38 y10f ffa fs7 fc0 sc0 lsa6">I<span class="ff1 fs5 fc1 ls0 ws23 v1">Assim como anterio<span class="_0 blank"></span>rmente, o número de vetores é igual à dimensão do</span></div><div class="t m0 x2d h1e y110 ff1 fs5 fc1 sc0 ls0 ws24">espaço em que eles são repr<span class="_0 blank"></span>esentados.<span class="_3 blank"> </span>Aplicando o teo<span class="_0 blank"></span>rema anterior,</div><div class="t m0 x2d h1e y111 ff1 fs5 fc1 sc0 ls0 ws1f">escreva:</div><div class="t m0 x37 h4b y112 ff3 fs5 fc1 sc0 lsf5">c<span class="ff1 fs2 lsf6 v1">1</span><span class="ff2 lsf7 v15">\ue012</span><span class="ff1 ls0 vd">2</span></div><div class="t m0 x1d h4c y113 ff1 fs5 fc1 sc0 lsf8">7<span class="ff2 lsf9 v11">\ue013</span><span class="ff4 fs6 ls63 vd">+</span><span class="ff3 ls6f vd">c</span><span class="fs2 lsfa v6">2</span><span class="ff2 lsfb v11">\ue012</span><span class="ls0 v16">1</span></div><div class="t m0 x63 h4c y114 ff1 fs5 fc1 sc0 lsf8">8<span class="ff2 lsfc v11">\ue013</span><span class="ff4 fs6 lsbd vd">=</span><span class="ff2 lsf7 v11">\ue012</span><span class="ls0 v16">0</span></div><div class="t m0 x64 h4c y115 ff1 fs5 fc1 sc0 lsfd">0<span class="ff2 lsfe v11">\ue013</span><span class="ff5 fs6 lsff vd">!</span><span class="ff2 lsf7 v11">\ue012</span><span class="ls0 ws4a v16">2 1</span></div><div class="t m0 x65 h4c y116 ff1 fs5 fc1 sc0 ls0 ws68">7<span class="_d blank"> </span>8 <span class="ff2 ws69 v11">\ue013 \ue012<span class="_e blank"> </span></span><span class="ff3 ls5e v16">c</span><span class="fs2 v17">1</span></div><div class="t m0 x48 h4d y117 ff3 fs5 fc1 sc0 ls5e">c<span class="ff1 fs2 ls100 v1">2</span><span class="ff2 ls101 v11">\ue013</span><span class="ff4 fs6 ls102 vd">=</span><span class="ff2 lsfb v11">\ue012</span><span class="ff1 ls0 v16">0</span></div><div class="t m0 x66 h4c y118 ff1 fs5 fc1 sc0 ls103">0<span class="ff2 ls0 v11">\ue013</span></div><div class="t m0 x36 h47 y119 ffa fs7 fc0 sc0 lsa6">I<span class="ff1 fs5 fc1 ls0 ws1e v1">O determinante da matriz de co<span class="_5 blank"> </span>e\u2026cientes é:</span></div><div class="t m0 x67 h39 y11a ff2 fs5 fc1 sc0 ls0">\ue00c</div><div class="t m0 x67 h39 y11b ff2 fs5 fc1 sc0 ls0">\ue00c</div><div class="t m0 x67 h39 y11c ff2 fs5 fc1 sc0 ls0">\ue00c</div><div class="t m0 x67 h39 y11d ff2 fs5 fc1 sc0 ls0">\ue00c</div><div class="t m0 x60 h1e y11e ff1 fs5 fc1 sc0 ls0 ws4a">2 1</div><div class="t m0 x60 h4e y11f ff1 fs5 fc1 sc0 ls0 ws65">7<span class="_d blank"> </span>8 <span class="ff2 v18">\ue00c</span></div><div class="t m0 x4c h39 y120 ff2 fs5 fc1 sc0 ls0">\ue00c</div><div class="t m0 x4c h39 y121 ff2 fs5 fc1 sc0 ls0">\ue00c</div><div class="t m0 x4c h4f y122 ff2 fs5 fc1 sc0 ls104">\ue00c<span class="ff4 fs6 ls6d vf">=</span><span class="ff1 ls0 ws1f vf">2<span class="ff7">.<span class="ff1 ls105">8<span class="ff5 fs6 ls106">\ue000</span></span></span>7<span class="ff7">.</span><span class="ls107">1<span class="ff4 fs6 ls67">=</span></span>9<span class="ff7">,</span></span></div><div class="t m0 x2d h50 y123 ff1 fs5 fc1 sc0 ls0 ws42">i.e.<span class="_6 blank"> </span>o<span class="_7 blank"> </span>sistema<span class="_7 blank"> </span>admite<span class="_7 blank"> </span>ap enas<span class="_7 blank"> </span>a<span class="_7 blank"> </span><span class="fc2 ws28 v0">solução trivial<span class="fc1 ls108">,<span class="ff3 ls5e">c</span><span class="fs2 ls70 v1">1</span><span class="ff4 fs6 ls6d">=</span><span class="ls0 ws6a">0 e <span class="ff3 ls5e">c</span><span class="fs2 ls70 v1">2</span><span class="ff4 fs6 ls67">=</span>0.</span></span></span></div><div class="t m0 x2d h51 y124 ff1 fs5 fc1 sc0 ls0 ws24">P<span class="_0 blank"></span>ortanto, os veto<span class="_0 blank"></span>res são <span class="fc2 ws26 v0">linearmente<span class="_7 blank"> </span>independentes<span class="fc1">.</span></span></div><div class="t m0 x59 h5 y32 ff1 fs2 fc1 sc0 ls0 ws3e">16</div></div><div class="c x0 y7 w2 h2"><div class="t m0 x7 h3 y33 ff1 fs0 fc0 sc0 ls0 ws3f">V<span class="_4 blank"></span>eri<span class="_0 blank"></span>\u2026ca<span class="_0 blank"></span>ndo<span class="_6 blank"> </span>In<span class="_0 blank"></span>depen<span class="_0 blank"></span>dên<span class="_0 blank"></span>cia<span class="_6 blank"> </span>Li<span class="_0 blank"></span>nea<span class="_4 blank"></span>r:<span class="_1 blank"> </span>E<span class="_0 blank"></span>xe<span class="_0 blank"></span>mp<span class="_0 blank"></span>los</div><div class="t m0 x8 h23 y125 ff1 fs1 fc2 sc0 ls0 ws7">Exemplo 3:<span class="_3 blank"> </span><span class="fc1 ws61">V<span class="_0 blank"></span>etores <span class="ff6 ws3">v</span><span class="fs2 ls9f v1">1</span><span class="ff4 fs4 lsee">=<span class="lsef v0">(</span></span><span class="ws3">2<span class="ff7 lsf0">,</span><span class="lsf1">7<span class="ff4 fs4 lsc9 v0">)</span><span class="ls109">,</span></span><span class="ff6">v</span><span class="fs2 ls10a v1">2</span><span class="ff4 fs4 ls14">=<span class="lsc9 v0">(</span></span>1<span class="ff7 ls10b">,</span><span class="ls10c">8<span class="ff4 fs4 ls10d v0">)</span><span class="ls9a">e</span></span><span class="ff6">v</span><span class="fs2 ls10e v1">3</span><span class="ff4 fs4 ls14">=<span class="lsc9 v0">(</span></span>1<span class="ff7 ls10f">,</span><span class="ls110">0<span class="ff4 fs4 ls10d v0">)</span></span><span class="ws56">em <span class="ff8 ls3e">R</span><span class="fs2 ls16 v7">2</span><span class="ff7">.</span></span></span></span></div><div class="t m0 x36 h38 y126 ffa fs7 fc0 sc0 lsa6">I<span class="ff1 fs5 fc1 ls0 ws28 v1">Neste caso, o número de veto<span class="_0 blank"></span>res é <span class="fc2 ws6b">maior </span>do que dimensão do espaço,</span></div><div class="t m0 x2d h20 y127 ff1 fs5 fc1 sc0 ls0 ws6c">i.e. <span class="ff3 ls111">k<span class="ff4 fs6 ls6d">=</span></span><span class="lsa7">3<span class="ffc fs6 ls6d">></span><span class="ff3 ls112">n<span class="ff4 fs6 ls113">=</span></span></span>2.</div><div class="t m0 x36 h3c y128 ffa fs7 fc0 sc0 lsa6">I<span class="ff1 fs5 fc1 ls0 ws28 v1">Ainda assim, p<span class="_5 blank"> </span>odemos aplicar o teo<span class="_4 blank"></span>rema<span class="_2 blank"> </span>anterio<span class="_0 blank"></span>r, expressando os</span></div><div class="t m0 x2d h52 y129 ff1 fs5 fc1 sc0 ls0 ws6d">veto<span class="_0 blank"></span>res no <span class="ff8 ls51">R</span><span class="fs2 ls16 v7">3</span><span class="ws28">, i.e.<span class="_3 blank"> </span><span class="ff6 ls114">v</span><span class="ff5 fs7 v19">0</span></span></div><div class="t m0 x20 h53 y12a ff1 fs2 fc1 sc0 ls70">1<span class="ff4 fs6 ls102 vf">=<span class="ls115 v0">(</span></span><span class="fs5 ls0 ws1f vf">2<span class="ff7 ls116">,</span>7<span class="ff7 ls116">,</span><span class="ls117">0<span class="ff4 fs6 ls115 v0">)</span><span class="ls108">,<span class="ff6 ls114">v</span></span></span><span class="ff5 fs7 v19">0</span></span></div><div class="t m0 x33 h53 y12a ff1 fs2 fc1 sc0 ls118">2<span class="ff4 fs6 lsbd vf">=<span class="ls119 v0">(</span></span><span class="fs5 ls0 ws1f vf">1<span class="ff7 ls11a">,</span>8<span class="ff7 ls11a">,</span><span class="ls11b">0<span class="ff4 fs6 ls11c v0">)</span><span class="lsc1">e<span class="ff6 ls114">v</span></span></span><span class="ff5 fs7 v19">0</span></span></div><div class="t m0 x68 h54 y12a ff1 fs2 fc1 sc0 ls11d">3<span class="ff4 fs6 ls102 vf">=<span class="ls115 v0">(</span></span><span class="fs5 ls0 ws1f vf">1<span class="ff7 ls11e">,<span class="ff1 ls55">0</span><span class="ls11f">,<span class="ff1 ls11b">0<span class="ff4 fs6 ls120 v0">)</span></span><span class="ls0">.</span></span></span></span></div><div class="t m0 x36 h38 y12b ffa fs7 fc0 sc0 lsa6">I<span class="ff1 fs5 fc1 ls0 ws6a v1">Desta fo<span class="_0 blank"></span>rma, obtemos o seguinte sistema:</span></div><div class="t m0 x69 h39 y12c ff2 fs5 fc1 sc0 ls0">0</div><div class="t m0 x69 h39 y12d ff2 fs5 fc1 sc0 ls0">@</div><div class="t m0 xf h1e y12e ff1 fs5 fc1 sc0 ls0 ws4a">2 1 1</div><div class="t m0 xf h1e y12f ff1 fs5 fc1 sc0 ls0 ws4a">7 8 0</div><div class="t m0 xf h1e y130 ff1 fs5 fc1 sc0 ls0 ws4a">0 0 0</div><div class="t m0 x6a h39 y131 ff2 fs5 fc1 sc0 ls0">1</div><div class="t m0 x6a h55 y132 ff2 fs5 fc1 sc0 ls121">A<span class="ls0 v1a">0</span></div><div class="t m0 x6b h39 y132 ff2 fs5 fc1 sc0 ls0">@</div><div class="t m0 x25 h1e y133 ff3 fs5 fc1 sc0 ls5b">c<span class="ff1 fs2 ls0 v1">1</span></div><div class="t m0 x25 h1e y134 ff3 fs5 fc1 sc0 ls6f">c<span class="ff1 fs2 ls0 v1">2</span></div><div class="t m0 x25 h1e y135 ff3 fs5 fc1 sc0 ls6f">c<span class="ff1 fs2 ls0 v1">3</span></div><div class="t m0 x6c h39 y136 ff2 fs5 fc1 sc0 ls0">1</div><div class="t m0 x6c h56 y137 ff2 fs5 fc1 sc0 ls122">A<span class="ff4 fs6 ls123 v3">=</span><span class="ls0 v1a">0</span></div><div class="t m0 x53 h39 y138 ff2 fs5 fc1 sc0 ls0">@</div><div class="t m0 x6d h1e y139 ff1 fs5 fc1 sc0 ls0">0</div><div class="t m0 x6d h1e y134 ff1 fs5 fc1 sc0 ls0">0</div><div class="t m0 x6d h1e y13a ff1 fs5 fc1 sc0 ls0">0</div><div class="t m0 x18 h39 y13b ff2 fs5 fc1 sc0 ls0">1</div><div class="t m0 x18 h39 y13c ff2 fs5 fc1 sc0 ls0">A</div><div class="t m0 x2d h1e y13d ff1 fs5 fc1 sc0 ls0 ws28">Note que o determinante da matriz de co<span class="_5 blank"> </span>e\u2026cientes é zero, pois ela</div><div class="t m0 x2d h1e y13e ff1 fs5 fc1 sc0 ls0 ws23">p<span class="_5 blank"> </span>ossui uma linh<span class="_0 blank"></span>a com entradas to<span class="_5 blank"> </span>das nulas.<span class="_6 blank"> </span>Assim, o sistema admite</div><div class="t m0 x2d h57 y13f ff1 fs5 fc1 sc0 ls0 ws6e">uma <span class="fc2 ws28 v0">solução não-nula <span class="fc1 ws40">e os veto<span class="_0 blank"></span>res são <span class="fc2 ws26">lineramente<span class="_7 blank"> </span>dep endentes</span>.</span></span></div><div class="t m0 x59 h5 y4e ff1 fs2 fc1 sc0 ls0 ws3e">17</div></div><div class="c x0 ya w2 h2"><div class="t m0 x7 h3 yb ff1 fs0 fc0 sc0 ls0 ws3f">V<span class="_4 blank"></span>eri<span class="_0 blank"></span>\u2026ca<span class="_0 blank"></span>ndo<span class="_6 blank"> </span>In<span class="_0 blank"></span>depen<span class="_0 blank"></span>dên<span class="_0 blank"></span>cia<span class="_6 blank"> </span>Li<span class="_0 blank"></span>nea<span class="_4 blank"></span>r:<span class="_1 blank"> </span>E<span class="_0 blank"></span>xe<span class="_0 blank"></span>mp<span class="_0 blank"></span>los</div><div class="t m0 x8 h4 y140 ff1 fs1 fc1 sc0 ls0 ws3">(Cont.)</div><div class="t m0 x36 h58 y141 ffa fs7 fc0 sc0 lsa6">I<span class="ff1 fs5 fc1 ls0 ws28 v1">Em pa<span class="_0 blank"></span>rticular, note que po<span class="_5 blank"> </span>demos exp<span class="_0 blank"></span>ressar o veto<span class="_0 blank"></span>r <span class="ff4 fs6 ls124 v0">(</span><span class="ws1f">2<span class="ff7 ls125">,</span><span class="ls126">7<span class="ff4 fs6 ls127 v0">)</span></span>como</span></span></div><div class="t m0 x2d h1e y142 ff1 fs5 fc1 sc0 ls0 ws66">combinção linea<span class="_0 blank"></span>r dos demais:</div><div class="t m0 x44 h1e y143 ff1 fs5 fc1 sc0 ls0">9</div><div class="t m0 x44 h59 y144 ff1 fs5 fc1 sc0 ls128">8<span class="ff2 lsfb v1b">\ue012</span><span class="ls0 v1c">1</span></div><div class="t m0 x60 h4c y145 ff1 fs5 fc1 sc0 ls103">0<span class="ff2 ls129 v11">\ue013</span><span class="ff4 fs6 ls12a vd">+</span><span class="ls0 v1d">7</span></div><div class="t m0 x32 h59 y144 ff1 fs5 fc1 sc0 ls12b">8<span class="ff2 ls12c v1b">\ue012</span><span class="ls0 v1c">1</span></div><div class="t m0 x25 h4c y145 ff1 fs5 fc1 sc0 ls103">8<span class="ff2 ls101 v11">\ue013</span><span class="ff4 fs6 ls123 vd">=</span><span class="ff2 lsfb v11">\ue012</span><span class="ls0 v16">2</span></div><div class="t m0 x6e h4c y146 ff1 fs5 fc1 sc0 ls103">7<span class="ff2 ls0 v11">\ue013</span></div><div class="t m0 x59 h5 y147 ff1 fs2 fc1 sc0 ls0 ws3e">18</div></div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div> <div id="pf7" class="pf w0 h0" data-page-no="7"><div class="pc pc7 w0 h0"><img fetchpriority="low" loading="lazy" class="bi x0 y0 w1 h1" alt="" src="https://files.passeidireto.com/e1597c04-9ff2-4637-afbf-78fed828f3b2/bg7.png"><div class="c x0 y1 w2 h2"><div class="t m0 x7 h3 y15 ff1 fs0 fc0 sc0 ls0 wsd">In<span class="_0 blank"></span>depen<span class="_0 blank"></span>dên<span class="_0 blank"></span>cia<span class="_6 blank"> </span>Li<span class="_0 blank"></span>nea<span class="_4 blank"></span>r</div><div class="t m0 x8 h1a y148 ff1 fs1 fc2 sc0 ls0 ws4f">T<span class="_4 blank"></span>eo<span class="_0 blank"></span>rema: <span class="fc1 ws6f">Se <span class="ff3 ls12d">k<span class="ffc fs4 ls1c">></span><span class="ls12e">n</span></span><span class="ws2">, qualquer conjunto de <span class="ff3 lsdd">k</span><span class="ws70">veto<span class="_0 blank"></span>res em <span class="ff8 ls45">R<span class="ff3 fs2 lsc5 v7">n</span></span>é</span></span></span></div><div class="t m0 x8 h4 y149 ff1 fs1 fc1 sc0 ls0 ws12">linea<span class="_0 blank"></span>rmente<span class="_2 blank"> </span>dep endente.</div><div class="t m0 x59 h5 y14a ff1 fs2 fc1 sc0 ls0 ws3e">19</div></div><div class="c x0 y7 w2 h2"><div class="t m0 x6f h6 y14b ff1 fs3 fc1 sc0 ls0 ws71">E<span class="_4 blank"></span>x<span class="_0 blank"></span>tr<span class="_4 blank"></span>a<span class="_0 blank"></span>:<span class="_f blank"> </span>U<span class="_4 blank"></span>s<span class="_0 blank"></span>a<span class="_0 blank"></span>n<span class="_0 blank"></span>d<span class="_4 blank"></span>o o E<span class="_0 blank"></span>x<span class="_0 blank"></span>c<span class="_0 blank"></span>e<span class="_0 blank"></span>l</div><div class="t m0 x59 h5 y9 ff1 fs2 fc1 sc0 ls0 ws3e">20</div></div><div class="c x0 ya w2 h2"><div class="t m0 x7 h3 yb ff1 fs0 fc0 sc0 ls0 ws72">Ex<span class="_0 blank"></span>cel<span class="_0 blank"></span>: "M<span class="_0 blank"></span>A<span class="_10 blank"></span>TR<span class="_0 blank"></span>IZ<span class="_0 blank"></span>.MU<span class="_4 blank"></span>L<span class="_10 blank"></span>T"</div><div class="t m0 x8 h4 y14c ff1 fs1 fc1 sc0 ls0 ws8">Usando o excel pa<span class="_0 blank"></span>ra multiplicar matrizes:</div><div class="t m0 x8 h4 y14d ff1 fs1 fc1 sc0 ls0 ws3">Exemplo:</div><div class="t m0 x36 h5a y14e ffa fs7 fc0 sc0 lsa6">I<span class="ff1 fs5 fc1 ls0 ws24 v1">Insira as matrizes <span class="ff3 lsb4">A<span class="ff1 lsc1">e</span><span class="ls12f">B</span></span><span class="ws28">resp<span class="_5 blank"> </span>ectivamente no espaço entre as células</span></span></div><div class="t m0 x2d h1e y14f ff1 fs5 fc1 sc0 ls0 ws28">C2:E3 e C5:D7<span class="ff7">.</span></div><div class="t m0 x36 h5b y150 ffa fs7 fc0 sc0 lsa6">I<span class="ff1 fs5 fc1 ls0 ws73 v1">Selecione as células C9:D10 e p<span class="_0 blank"></span>ressione F2 para po<span class="_5 blank"> </span>der inserir a fórmula.</span></div><div class="t m0 x36 h5a y151 ffa fs7 fc0 sc0 lsa6">I<span class="ff1 fs5 fc1 ls0 ws74 v1">Digite <span class="ff3 ws75">=MA<span class="_4 blank"></span>TRIZ.MUL<span class="_4 blank"></span>T(C <span class="ff1 ws1f">2</span><span class="ws76">:E <span class="ff1 ws1f">3<span class="ff7">;</span></span><span class="ls130">C</span><span class="ff1 ws1f">5</span><span class="ws77">:D <span class="ff1 ws1f">7</span><span class="ls131">)</span><span class="ff1 ws28">e p<span class="_0 blank"></span>resione Ctrl+Shift+Enter.</span></span></span></span></span></div><div class="t m0 x59 h5 y14 ff1 fs2 fc1 sc0 ls0 ws3e">21</div></div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div> <div id="pf8" class="pf w0 h0" data-page-no="8"><div class="pc pc8 w0 h0"><img fetchpriority="low" loading="lazy" class="bi x0 y152 w1 h5c" alt="" src="https://files.passeidireto.com/e1597c04-9ff2-4637-afbf-78fed828f3b2/bg8.png"><div class="c x0 y1 w2 h2"><div class="t m0 x7 h3 y15 ff1 fs0 fc0 sc0 ls0 ws72">Ex<span class="_0 blank"></span>cel<span class="_0 blank"></span>: "M<span class="_0 blank"></span>A<span class="_10 blank"></span>TR<span class="_0 blank"></span>IZ<span class="_0 blank"></span>.IN<span class="_0 blank"></span>VE<span class="_0 blank"></span>RS<span class="_0 blank"></span>O"</div><div class="t m0 x8 h4 y153 ff1 fs1 fc1 sc0 ls0 ws7">Usando o excel pa<span class="_0 blank"></span>ra inverter matrizes:</div><div class="t m0 x8 h4 y154 ff1 fs1 fc1 sc0 ls0 ws3">Exemplo:</div><div class="t m0 x36 h38 y155 ffa fs7 fc0 sc0 lsa6">I<span class="ff1 fs5 fc1 ls0 ws66 v1">Insira a matriz <span class="ff3 lsb4">A</span><span class="ws1e">nas células C2:E4<span class="ff7">.</span></span></span></div><div class="t m0 x36 h3c y156 ffa fs7 fc0 sc0 lsa6">I<span class="ff1 fs5 fc1 ls0 ws24 v1">Selecione as células C6:E8 e p<span class="_0 blank"></span>ressione F2 para po<span class="_5 blank"> </span>der inserir a fórmula.</span></div><div class="t m0 x36 h38 y157 ffa fs7 fc0 sc0 lsa6">I<span class="ff1 fs5 fc1 ls0 ws74 v1">Digite <span class="ff3 ws78">=MA<span class="_4 blank"></span>TRIZ.INVERSO(C <span class="ff1 ws1f">2</span><span class="ws76">:E <span class="ff1 ws1f">4</span><span class="ls131">)</span><span class="ff1 ws6a">e p<span class="_0 blank"></span>resione Ctrl+Shift+Enter.</span></span></span></span></div><div class="t m0 x59 h5 y32 ff1 fs2 fc1 sc0 ls0 ws3e">22</div></div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div>
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